Elementary algebra
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Elementary algebra, also known as high school algebra or college algebra,<ref>Pierce, R., College Algebra, Maths is Fun, accessed 28 August 2023</ref> encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers,<ref>H.E. Slaught and N.J. Lennes, Elementary algebra, Publ. Allyn and Bacon, 1915, page 1 (republished by Forgotten Books)</ref> whilst algebra introduces variables (quantities without fixed values).<ref>Lewis Hirsch, Arthur Goodman, Understanding Elementary Algebra With Geometry: A Course for College Students, Publisher: Cengage Learning, 2005, Template:ISBN, 9780534999728, 654 pages, page 2</ref>
This use of variables entails use of algebraic notation and an understanding of the general rules of the operations introduced in arithmetic: addition, subtraction, multiplication, division, etc. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.
It is typically taught to secondary school students and at introductory college level in the United States,<ref name=leff /> and builds on their understanding of arithmetic. The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Many quantitative relationships in science and mathematics are expressed as algebraic equations.
Algebraic operationsEdit
Algebraic notationEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Algebraic notation describes the rules and conventions for writing mathematical expressions, as well as the terminology used for talking about parts of expressions. For example, the expression <math style="margin-bottom:8px">3x^2 - 2xy + c</math> has the following components:
1 – power (exponent)
2 – coefficient
3 – term
4 – operator
5 – constant term
<math>c</math> – constant
<math>x</math> <math>y</math> – variables
A coefficient is a numerical value, or letter representing a numerical constant, that multiplies a variable (the operator is omitted). A term is an addend or a summand, a group of coefficients, variables, constants and exponents that may be separated from the other terms by the plus and minus operators.<ref>Richard N. Aufmann, Joanne Lockwood, Introductory Algebra: An Applied Approach, Publisher Cengage Learning, 2010, Template:ISBN, 9781439046043, page 78</ref> Letters represent variables and constants. By convention, letters at the beginning of the alphabet (e.g. <math>a, b, c</math>) are typically used to represent constants, and those toward the end of the alphabet (e.g. <math>x, y</math> and Template:Mvar) are used to represent variables.<ref>William L. Hosch (editor), The Britannica Guide to Algebra and Trigonometry, Britannica Educational Publishing, The Rosen Publishing Group, 2010, Template:ISBN, 9781615302192, page 71</ref> They are usually printed in italics.<ref>James E. Gentle, Numerical Linear Algebra for Applications in Statistics, Publisher: Springer, 1998, Template:ISBN, 9780387985428, 221 pages, [James E. Gentle page 184]</ref>
Algebraic operations work in the same way as arithmetic operations,<ref>Horatio Nelson Robinson, New elementary algebra: containing the rudiments of science for schools and academies, Ivison, Phinney, Blakeman, & Co., 1866, page 7</ref> such as addition, subtraction, multiplication, division and exponentiation,<ref>Ron Larson, Robert Hostetler, Bruce H. Edwards, Algebra And Trigonometry: A Graphing Approach, Publisher: Cengage Learning, 2007, Template:ISBN, 9780618851959, 1114 pages, page 6</ref> and are applied to algebraic variables and terms. Multiplication symbols are usually omitted, and implied when there is no space between two variables or terms, or when a coefficient is used. For example, <math style="margin-bottom:8px">3 \times x^2</math> is written as <math style="margin-bottom:8px">3x^2</math>, and <math>2 \times x \times y</math> may be written <math>2xy</math>.<ref>Sin Kwai Meng, Chip Wai Lung, Ng Song Beng, "Algebraic notation", in Mathematics Matters Secondary 1 Express Textbook, Publisher Panpac Education Pte Ltd, Template:ISBN, 9789812738820, page 68</ref>
Usually terms with the highest power (exponent), are written on the left, for example, <math style="margin-bottom:8px">x^2</math> is written to the left of Template:Mvar. When a coefficient is one, it is usually omitted (e.g. <math style="margin-bottom:8px">1x^2</math> is written <math style="margin-bottom:8px">x^2</math>).<ref>David Alan Herzog, Teach Yourself Visually Algebra, Publisher John Wiley & Sons, 2008, Template:ISBN, 9780470185599, 304 pages, page 72</ref> Likewise when the exponent (power) is one, (e.g. <math style="margin-bottom:8px">3x^1</math> is written <math style="margin-bottom:8px">3x</math>).<ref>John C. Peterson, Technical Mathematics With Calculus, Publisher Cengage Learning, 2003, Template:ISBN, 9780766861893, 1613 pages, page 31</ref> When the exponent is zero, the result is always 1 (e.g. <math style="margin-bottom:8px">x^0</math> is always rewritten to Template:Mvar).<ref>Jerome E. Kaufmann, Karen L. Schwitters, Algebra for College Students, Publisher Cengage Learning, 2010, Template:ISBN, 9780538733540, 803 pages, page 222</ref> However <math>0^0</math>, being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents.
Alternative notationEdit
Other types of notation are used in algebraic expressions when the required formatting is not available, or can not be implied, such as where only letters and symbols are available. As an illustration of this, while exponents are usually formatted using superscripts, e.g., <math style="margin-bottom:8px">x^2</math>, in plain text, and in the TeX mark-up language, the caret symbol Template:Char represents exponentiation, so <math style="margin-bottom:8px">x^2</math> is written as "x^2".<ref>Ramesh Bangia, Dictionary of Information Technology, Publisher Laxmi Publications, Ltd., 2010, Template:ISBN, 9789380298153, page 212</ref><ref>George Grätzer, First Steps in LaTeX, Publisher Springer, 1999, Template:ISBN, 9780817641320, page 17</ref> This also applies to some programming languages such as Lua. In programming languages such as Ada,<ref>S. Tucker Taft, Robert A. Duff, Randall L. Brukardt, Erhard Ploedereder, Pascal Leroy, Ada 2005 Reference Manual, Volume 4348 of Lecture Notes in Computer Science, Publisher Springer, 2007, Template:ISBN, 9783540693352, page 13</ref> Fortran,<ref>C. Xavier, Fortran 77 And Numerical Methods, Publisher New Age International, 1994, Template:ISBN, 9788122406702, page 20</ref> Perl,<ref>Randal Schwartz, Brian Foy, Tom Phoenix, Learning Perl, Publisher O'Reilly Media, Inc., 2011, Template:ISBN, 9781449313142, page 24</ref> Python<ref>Matthew A. Telles, Python Power!: The Comprehensive Guide, Publisher Course Technology PTR, 2008, Template:ISBN, 9781598631586, page 46</ref> and Ruby,<ref>Kevin C. Baird, Ruby by Example: Concepts and Code, Publisher No Starch Press, 2007, Template:ISBN, 9781593271480, page 72</ref> a double asterisk is used, so <math style="margin-bottom:8px">x^2</math> is written as "x**2". Many programming languages and calculators use a single asterisk to represent the multiplication symbol,<ref>William P. Berlinghoff, Fernando Q. Gouvêa, Math through the Ages: A Gentle History for Teachers and Others, Publisher MAA, 2004, Template:ISBN, 9780883857366, page 75</ref> and it must be explicitly used, for example, <math style="margin-bottom:8px">3x</math> is written "3*x".
ConceptsEdit
VariablesEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Elementary algebra builds on and extends arithmetic<ref>Thomas Sonnabend, Mathematics for Teachers: An Interactive Approach for Grades K-8, Publisher: Cengage Learning, 2009, Template:ISBN, 9780495561668, 759 pages, page xvii</ref> by introducing letters called variables to represent general (non-specified) numbers. This is useful for several reasons.
- Variables may represent numbers whose values are not yet known. For example, if the temperature of the current day, C, is 20 degrees higher than the temperature of the previous day, P, then the problem can be described algebraically as <math>C = P + 20</math>.<ref>Lewis Hirsch, Arthur Goodman, Understanding Elementary Algebra With Geometry: A Course for College Students, Publisher: Cengage Learning, 2005, Template:ISBN, 9780534999728, 654 pages, page 48</ref>
- Variables allow one to describe general problems,<ref name=leff>Lawrence S. Leff, College Algebra: Barron's Ez-101 Study Keys, Publisher: Barron's Educational Series, 2005, Template:ISBN, 9780764129148, 230 pages, page 2</ref> without specifying the values of the quantities that are involved. For example, it can be stated specifically that 5 minutes is equivalent to <math>60 \times 5 = 300</math> seconds. A more general (algebraic) description may state that the number of seconds, <math>s = 60 \times m</math>, where m is the number of minutes.
- Variables allow one to describe mathematical relationships between quantities that may vary.<ref>Ron Larson, Kimberly Nolting, Elementary Algebra, Publisher: Cengage Learning, 2009, Template:ISBN, 9780547102276, 622 pages, page 210</ref> For example, the relationship between the circumference, c, and diameter, d, of a circle is described by <math>\pi = c /d</math>.
- Variables allow one to describe some mathematical properties. For example, a basic property of addition is commutativity which states that the order of numbers being added together does not matter. Commutativity is stated algebraically as <math>(a + b) = (b + a)</math>.<ref>Charles P. McKeague, Elementary Algebra, Publisher: Cengage Learning, 2011, Template:ISBN, 9780840064219, 571 pages, page 49</ref>
Simplifying expressionsEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations (addition, subtraction, multiplication, division and exponentiation). For example,
- Added terms are simplified using coefficients. For example, <math>x + x + x</math> can be simplified as <math>3x</math> (where 3 is a numerical coefficient).
- Multiplied terms are simplified using exponents. For example, <math>x \times x \times x</math> is represented as <math>x^3</math>
- Like terms are added together,<ref>Andrew Marx, Shortcut Algebra I: A Quick and Easy Way to Increase Your Algebra I Knowledge and Test Scores, Publisher Kaplan Publishing, 2007, Template:ISBN, 9781419552885, 288 pages, page 51</ref> for example, <math>2x^2 + 3ab - x^2 + ab</math> is written as <math>x^2 + 4ab</math>, because the terms containing <math>x^2</math> are added together, and the terms containing <math>ab</math> are added together.
- Brackets can be "multiplied out", using the distributive property. For example, <math>x (2x + 3)</math> can be written as <math>(x \times 2x) + (x \times 3)</math> which can be written as <math>2x^2 + 3x</math>
- Expressions can be factored. For example, <math>6x^5 + 3x^2</math>, by dividing both terms by the common factor, <math>3x^2</math> can be written as <math>3x^2 (2x^3 + 1)</math>
EquationsEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} An equation states that two expressions are equal using the symbol for equality, = (the equals sign).<ref>Mark Clark, Cynthia Anfinson, Beginning Algebra: Connecting Concepts Through Applications, Publisher Cengage Learning, 2011, Template:ISBN, 9780534419387, 793 pages, page 134</ref> One of the best-known equations describes Pythagoras' law relating the length of the sides of a right angle triangle:<ref>Alan S. Tussy, R. David Gustafson, Elementary and Intermediate Algebra, Publisher Cengage Learning, 2012, Template:ISBN, 9781111567682, 1163 pages, page 493</ref>
- <math>c^2 = a^2 + b^2</math>
This equation states that <math>c^2</math>, representing the square of the length of the side that is the hypotenuse, the side opposite the right angle, is equal to the sum (addition) of the squares of the other two sides whose lengths are represented by Template:Mvar and Template:Mvar.
An equation is the claim that two expressions have the same value and are equal. Some equations are true for all values of the involved variables (such as <math>a + b = b + a</math>); such equations are called identities. Conditional equations are true for only some values of the involved variables, e.g. <math>x^2 - 1 = 8</math> is true only for <math>x = 3</math> and <math>x = -3</math>. The values of the variables which make the equation true are the solutions of the equation and can be found through equation solving.
Another type of equation is inequality. Inequalities are used to show that one side of the equation is greater, or less, than the other. The symbols used for this are: <math> a > b </math> where <math> > </math> represents 'greater than', and <math> a < b </math> where <math> < </math> represents 'less than'. Just like standard equality equations, numbers can be added, subtracted, multiplied or divided. The only exception is that when multiplying or dividing by a negative number, the inequality symbol must be flipped.
Properties of equalityEdit
By definition, equality is an equivalence relation, meaning it is reflexive (i.e. <math>b = b</math>), symmetric (i.e. if <math>a = b</math> then <math>b = a</math>), and transitive (i.e. if <math>a = b</math> and <math>b = c</math> then <math>a = c</math>).<ref>Douglas Downing, Algebra the Easy Way, Publisher Barron's Educational Series, 2003, Template:ISBN, 9780764119729, 392 pages, page 20</ref> It also satisfies the important property that if two symbols are used for equal things, then one symbol can be substituted for the other in any true statement about the first and the statement will remain true. This implies the following properties:
- if <math>a = b</math> and <math>c = d</math> then <math>a + c = b + d</math> and <math>ac = bd</math>;
- if <math>a = b</math> then <math>a + c = b + c</math> and <math>ac = bc</math>;
- more generally, for any function Template:Mvar, if <math>a=b</math> then <math>f(a) = f(b)</math>.
Properties of inequalityEdit
The relations less than <math> < </math> and greater than <math> > </math> have the property of transitivity:<ref>Ron Larson, Robert Hostetler, Intermediate Algebra, Publisher Cengage Learning, 2008, Template:ISBN, 9780618753529, 857 pages, page 96</ref>
- If <math>a < b</math> and <math>b < c</math> then <math>a < c</math>;
- If <math>a < b</math> and <math>c < d</math> then <math>a + c < b + d</math>;<ref>{{#invoke:citation/CS1|citation
|CitationClass=web }}</ref>
- If <math>a < b</math> and <math>c > 0</math> then <math>ac < bc</math>;
- If <math>a < b</math> and <math>c < 0</math> then <math>bc < ac</math>.
By reversing the inequation, <math> < </math> and <math> > </math> can be swapped,<ref>Chris Carter, Physics: Facts and Practice for A Level, Publisher Oxford University Press, 2001, Template:ISBN, 9780199147687, 144 pages, page 50</ref> for example:
- <math>a < b</math> is equivalent to <math>b > a</math>
SubstitutionEdit
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Substitution is replacing the terms in an expression to create a new expression. Substituting 3 for Template:Mvar in the expression Template:Math makes a new expression Template:Math with meaning Template:Math. Substituting the terms of a statement makes a new statement. When the original statement is true independently of the values of the terms, the statement created by substitutions is also true. Hence, definitions can be made in symbolic terms and interpreted through substitution: if <math>a^2:=a\times a</math> is meant as the definition of <math>a^2,</math> as the product of Template:Mvar with itself, substituting Template:Math for Template:Mvar informs the reader of this statement that <math>3^2</math> means Template:Math. Often it's not known whether the statement is true independently of the values of the terms. And, substitution allows one to derive restrictions on the possible values, or show what conditions the statement holds under. For example, taking the statement Template:Math, if Template:Mvar is substituted with Template:Math, this implies Template:Math, which is false, which implies that if Template:Math then Template:Mvar cannot be Template:Math.
If Template:Math and Template:Math are integers, rationals, or real numbers, then Template:Math implies Template:Math or Template:Math. Consider Template:Math. Then, substituting Template:Math for Template:Math and Template:Math for Template:Math, we learn Template:Math or Template:Math. Then we can substitute again, letting Template:Math and Template:Math, to show that if Template:Math then Template:Math or Template:Math. Therefore, if Template:Math, then Template:Math or (Template:Math or Template:Math), so Template:Math implies Template:Math or Template:Math or Template:Math.
If the original fact were stated as "Template:Math implies Template:Math or Template:Math", then when saying "consider Template:Math," we would have a conflict of terms when substituting. Yet the above logic is still valid to show that if Template:Math then Template:Math or Template:Math or Template:Math if, instead of letting Template:Math and Template:Math, one substitutes Template:Math for Template:Math and Template:Math for Template:Math (and with Template:Math, substituting Template:Math for Template:Math and Template:Math for Template:Math). This shows that substituting for the terms in a statement isn't always the same as letting the terms from the statement equal the substituted terms. In this situation it's clear that if we substitute an expression Template:Math into the Template:Math term of the original equation, the Template:Math substituted does not refer to the Template:Math in the statement "Template:Math implies Template:Math or Template:Math."
Solving algebraic equationsEdit
The following sections lay out examples of some of the types of algebraic equations that may be encountered.
Linear equations with one variableEdit
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Linear equations are so-called, because when they are plotted, they describe a straight line. The simplest equations to solve are linear equations that have only one variable. They contain only constant numbers and a single variable without an exponent. As an example, consider:
- Problem in words: If you double the age of a child and add 4, the resulting answer is 12. How old is the child?
- Equivalent equation: <math>2x + 4 = 12</math> where Template:Mvar represent the child's age
To solve this kind of equation, the technique is add, subtract, multiply, or divide both sides of the equation by the same number in order to isolate the variable on one side of the equation. Once the variable is isolated, the other side of the equation is the value of the variable.<ref>Template:Cite book</ref> This problem and its solution are as follows:
| 1. Equation to solve: | <math>2x + 4 = 12</math> |
| 2. Subtract 4 from both sides: | <math>2x + 4 - 4 = 12 - 4</math> |
| 3. This simplifies to: | <math>2x = 8</math> |
| 4. Divide both sides by 2: | <math>\frac{2x}{2} = \frac{8}{2}</math> |
| 5. This simplifies to the solution: | <math>x = 4</math> |
In words: the child is 4 years old.
The general form of a linear equation with one variable, can be written as: <math>ax+b=c</math>
Following the same procedure (i.e. subtract Template:Mvar from both sides, and then divide by Template:Mvar), the general solution is given by <math>x=\frac{c-b}{a}</math>
Linear equations with two variablesEdit
A linear equation with two variables has many (i.e. an infinite number of) solutions.<ref>Sinha, The Pearson Guide to Quantitative Aptitude for CAT 2/ePublisher: Pearson Education India, 2010, Template:ISBN, 9788131723661, 599 pages, page 195</ref> For example:
- Problem in words: A father is 22 years older than his son. How old are they?
- Equivalent equation: <math>y = x + 22</math> where Template:Mvar is the father's age, Template:Mvar is the son's age.
That cannot be worked out by itself. If the son's age was made known, then there would no longer be two unknowns (variables). The problem then becomes a linear equation with just one variable, that can be solved as described above.
To solve a linear equation with two variables (unknowns), requires two related equations. For example, if it was also revealed that:
- Problem in words
- In 10 years, the father will be twice as old as his son.
- Equivalent equation
- <math>\begin{align}
y + 10 &= 2 \times (x + 10)\\ y &= 2 \times (x + 10) - 10 && \text{Subtract 10 from both sides}\\ y &= 2x + 20 - 10 && \text{Multiple out brackets}\\ y &= 2x + 10 && \text{Simplify} \end{align}</math>
Now there are two related linear equations, each with two unknowns, which enables the production of a linear equation with just one variable, by subtracting one from the other (called the elimination method):<ref>Cynthia Y. Young, Precalculus, Publisher John Wiley & Sons, 2010, Template:ISBN, 9780471756842, 1175 pages, page 699</ref>
- <math>\begin{cases}
y = x + 22 & \text{First equation}\\ y = 2x + 10 & \text{Second equation} \end{cases}</math>
- <math>\begin{align}
&&&\text{Subtract the first equation from}\\ (y - y) &= (2x - x) +10 - 22 && \text{the second in order to remove } y\\ 0 &= x - 12 && \text{Simplify}\\ 12 &= x && \text{Add 12 to both sides}\\ x &= 12 && \text{Rearrange} \end{align}</math>
In other words, the son is aged 12, and since the father 22 years older, he must be 34. In 10 years, the son will be 22, and the father will be twice his age, 44. This problem is illustrated on the associated plot of the equations.
For other ways to solve this kind of equations, see below, System of linear equations.
Quadratic equationsEdit
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A quadratic equation is one which includes a term with an exponent of 2, for example, <math>x^2</math>,<ref>Mary Jane Sterling, Algebra II For Dummies, Publisher: John Wiley & Sons, 2006, Template:ISBN, 9780471775812, 384 pages, page 37</ref> and no term with higher exponent. The name derives from the Latin quadrus, meaning square.<ref>John T. Irwin, The Mystery to a Solution: Poe, Borges, and the Analytic Detective Story, Publisher JHU Press, 1996, Template:ISBN, 9780801854668, 512 pages, page 372</ref> In general, a quadratic equation can be expressed in the form <math>ax^2 + bx + c = 0</math>,<ref>Sharma/khattar, The Pearson Guide To Objective Mathematics For Engineering Entrance Examinations, 3/E, Publisher Pearson Education India, 2010, Template:ISBN, 9788131723630, 1248 pages, page 621</ref> where Template:Mvar is not zero (if it were zero, then the equation would not be quadratic but linear). Because of this a quadratic equation must contain the term <math>ax^2</math>, which is known as the quadratic term. Hence <math>a \neq 0</math>, and so we may divide by Template:Mvar and rearrange the equation into the standard form
- <math>x^2 + px + q = 0 </math>
where <math>p = \frac{b}{a}</math> and <math>q = \frac{c}{a}</math>. Solving this, by a process known as completing the square, leads to the quadratic formula
- <math>x=\frac{-b \pm \sqrt {b^2-4ac}}{2a},</math>
where the symbol "±" indicates that both
- <math> x=\frac{-b + \sqrt {b^2-4ac}}{2a}\quad\text{and}\quad x=\frac{-b - \sqrt {b^2-4ac}}{2a}</math>
are solutions of the quadratic equation.
Quadratic equations can also be solved using factorization (the reverse process of which is expansion, but for two linear terms is sometimes denoted foiling). As an example of factoring:
- <math>x^{2} + 3x - 10 = 0, </math>
which is the same thing as
- <math>(x + 5)(x - 2) = 0. </math>
It follows from the zero-product property that either <math>x = 2</math> or <math>x = -5</math> are the solutions, since precisely one of the factors must be equal to zero. All quadratic equations will have two solutions in the complex number system, but need not have any in the real number system. For example,
- <math>x^{2} + 1 = 0 </math>
has no real number solution since no real number squared equals −1. Sometimes a quadratic equation has a root of multiplicity 2, such as:
- <math>(x + 1)^2 = 0. </math>
For this equation, −1 is a root of multiplicity 2. This means −1 appears twice, since the equation can be rewritten in factored form as
- <math>[x-(-1)][x-(-1)]=0.</math>
Complex numbersEdit
All quadratic equations have exactly two solutions in complex numbers (but they may be equal to each other), a category that includes real numbers, imaginary numbers, and sums of real and imaginary numbers. Complex numbers first arise in the teaching of quadratic equations and the quadratic formula. For example, the quadratic equation
- <math>x^2+x+1=0</math>
has solutions
- <math>x=\frac{-1 + \sqrt{-3}}{2} \quad \quad \text{and} \quad \quad x=\frac{-1-\sqrt{-3}}{2}.</math>
Since <math>\sqrt{-3}</math> is not any real number, both of these solutions for x are complex numbers.
Exponential and logarithmic equationsEdit
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An exponential equation is one which has the form <math>a^x = b</math> for <math>a > 0</math>,<ref>Aven Choo, LMAN OL Additional Maths Revision Guide 3, Publisher Pearson Education South Asia, 2007, Template:ISBN, 9789810600013, page 105</ref> which has solution
- <math>x = \log_a b = \frac{\ln b}{\ln a}</math>
when <math>b > 0</math>. Elementary algebraic techniques are used to rewrite a given equation in the above way before arriving at the solution. For example, if
- <math>3 \cdot 2^{x - 1} + 1 = 10</math>
then, by subtracting 1 from both sides of the equation, and then dividing both sides by 3 we obtain
- <math>2^{x - 1} = 3</math>
whence
- <math>x - 1 = \log_2 3</math>
or
- <math>x = \log_2 3 + 1.</math>
A logarithmic equation is an equation of the form <math>log_a(x) = b</math> for <math>a > 0</math>, which has solution
- <math>x = a^b.</math>
For example, if
- <math>4\log_5(x - 3) - 2 = 6</math>
then, by adding 2 to both sides of the equation, followed by dividing both sides by 4, we get
- <math>\log_5(x - 3) = 2</math>
whence
- <math>x - 3 = 5^2 = 25</math>
from which we obtain
- <math>x = 28.</math>
Radical equationsEdit
Template:Image frame A radical equation is one that includes a radical sign, which includes square roots, <math>\sqrt{x},</math> cube roots, <math>\sqrt[3]{x}</math>, and nth roots, <math>\sqrt[n]{x}</math>. Recall that an nth root can be rewritten in exponential format, so that <math>\sqrt[n]{x}</math> is equivalent to <math>x^{\frac{1}{n}}</math>. Combined with regular exponents (powers), then <math>\sqrt[2]{x^3}</math> (the square root of Template:Mvar cubed), can be rewritten as <math>x^{\frac{3}{2}}</math>.<ref>John C. Peterson, Technical Mathematics With Calculus, Publisher Cengage Learning, 2003, Template:ISBN, 9780766861893, 1613 pages, page 525</ref> So a common form of a radical equation is <math> \sqrt[n]{x^m}=a</math> (equivalent to <math> x^\frac{m}{n}=a</math>) where Template:Mvar and Template:Mvar are integers. It has real solution(s):
| Template:Mvar is odd | Template:Mvar is even and <math>a \ge 0</math> |
Template:Mvar and Template:Mvar are even and <math>a<0</math> |
Template:Mvar is even, Template:Mvar is odd, and <math>a<0</math> |
|---|---|---|---|
| <math>x = \sqrt[n]{a^m}</math>
equivalently
|
<math>x = \pm \sqrt[n]{a^m}</math>
equivalently
|
<math>x=\pm \sqrt[n]{a^m}</math> | no real solution |
For example, if:
- <math>(x + 5)^{2/3} = 4</math>
then
- <math>\begin{align}
x + 5 & = \pm (\sqrt{4})^3,\\ x + 5 & = \pm 8,\\ x & = -5 \pm 8, \end{align}</math> and thus
- <math>x = 3 \quad \text{or}\quad x = -13</math>
System of linear equationsEdit
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There are different methods to solve a system of linear equations with two variables.
Elimination methodEdit
An example of solving a system of linear equations is by using the elimination method:
- <math>\begin{cases}4x + 2y&= 14 \\
2x - y&= 1.\end{cases} </math>
Multiplying the terms in the second equation by 2:
- <math>4x + 2y = 14 </math>
- <math>4x - 2y = 2. </math>
Adding the two equations together to get:
- <math>8x = 16 </math>
which simplifies to
- <math>x = 2. </math>
Since the fact that <math>x = 2</math> is known, it is then possible to deduce that <math>y = 3</math> by either of the original two equations (by using 2 instead of Template:Mvar ) The full solution to this problem is then
- <math>\begin{cases} x = 2 \\ y = 3. \end{cases}</math>
This is not the only way to solve this specific system; Template:Mvar could have been resolved before Template:Mvar.
Substitution methodEdit
Another way of solving the same system of linear equations is by substitution.
- <math>\begin{cases}4x + 2y &= 14
\\ 2x - y &= 1.\end{cases} </math>
An equivalent for Template:Mvar can be deduced by using one of the two equations. Using the second equation:
- <math>2x - y = 1 </math>
Subtracting <math>2x</math> from each side of the equation:
- <math>\begin{align}2x - 2x - y & = 1 - 2x \\
- y & = 1 - 2x \end{align}</math>
and multiplying by −1:
- <math> y = 2x - 1. </math>
Using this Template:Mvar value in the first equation in the original system:
- <math>\begin{align}4x + 2(2x - 1) &= 14\\
4x + 4x - 2 &= 14 \\ 8x - 2 &= 14 \end{align}</math>
Adding 2 on each side of the equation:
- <math>\begin{align}8x - 2 + 2 &= 14 + 2 \\
8x &= 16 \end{align}</math>
which simplifies to
- <math>x = 2 </math>
Using this value in one of the equations, the same solution as in the previous method is obtained.
- <math>\begin{cases} x = 2 \\ y = 3. \end{cases}</math>
This is not the only way to solve this specific system; in this case as well, Template:Mvar could have been solved before Template:Mvar.
Other types of systems of linear equationsEdit
Inconsistent systemsEdit
In the above example, a solution exists. However, there are also systems of equations which do not have any solution. Such a system is called inconsistent. An obvious example is
- <math>\begin{cases}\begin{align} x + y &= 1 \\
0x + 0y &= 2\,. \end{align} \end{cases}</math>
As 0≠2, the second equation in the system has no solution. Therefore, the system has no solution. However, not all inconsistent systems are recognized at first sight. As an example, consider the system
- <math>\begin{cases}\begin{align}4x + 2y &= 12 \\
-2x - y &= -4\,. \end{align}\end{cases}</math>
Multiplying by 2 both sides of the second equation, and adding it to the first one results in
- <math>0x+0y = 4 \,,</math>
which clearly has no solution.
Undetermined systemsEdit
There are also systems which have infinitely many solutions, in contrast to a system with a unique solution (meaning, a unique pair of values for Template:Mvar and Template:Mvar) For example:
- <math>\begin{cases}\begin{align}4x + 2y & = 12 \\
-2x - y & = -6 \end{align}\end{cases}</math>
Isolating Template:Mvar in the second equation:
- <math>y = -2x + 6 </math>
And using this value in the first equation in the system:
- <math>\begin{align}4x + 2(-2x + 6) = 12 \\
4x - 4x + 12 = 12 \\ 12 = 12 \end{align}</math>
The equality is true, but it does not provide a value for Template:Mvar. Indeed, one can easily verify (by just filling in some values of Template:Mvar) that for any Template:Mvar there is a solution as long as <math>y = -2x + 6</math>. There is an infinite number of solutions for this system.
Over- and underdetermined systemsEdit
Systems with more variables than the number of linear equations are called underdetermined. Such a system, if it has any solutions, does not have a unique one but rather an infinitude of them. An example of such a system is
- <math>\begin{cases}\begin{align}x + 2y & = 10\\
y - z & = 2 .\end{align}\end{cases}</math>
When trying to solve it, one is led to express some variables as functions of the other ones if any solutions exist, but cannot express all solutions numerically because there are an infinite number of them if there are any.
A system with a higher number of equations than variables is called overdetermined. If an overdetermined system has any solutions, necessarily some equations are linear combinations of the others.
See alsoEdit
- History of algebra
- Binary operation
- Gaussian elimination
- Mathematics education
- Number line
- Polynomial
- Cancelling out
- Tarski's high school algebra problem
ReferencesEdit
- Leonhard Euler, Elements of Algebra, 1770. English translation Tarquin Press, 2007, Template:ISBN, also online digitized editions<ref>Euler's Elements of Algebra Template:Webarchive</ref> 2006,<ref>{{#invoke:citation/CS1|citation
|CitationClass=web }}</ref> 1822.
- Charles Smith, A Treatise on Algebra, in Cornell University Library Historical Math Monographs.
- Redden, John. Elementary Algebra Template:Webarchive. Flat World Knowledge, 2011
External linksEdit
Template:Algebra Template:Areas of mathematics Template:Authority control