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Solve for a: 4 a + 2 a = 3 + 5 a 2 4 a + 2 a = 3 + 5 a 2 Add same side like terms first. 5 a + 2 = 5 a + 1 5 a + 2 5 a = 5 a + 1 5 a Subtract 5a from both sides . 10 a + 2 = 1 10 a + 2 2 = 1 2 Subtract 2 from both sides . 10 a = 1 10 a 10 = 1 10 Divide both sides by -10 . a = 1 10 The solution set is { 1 10 }

Simplifying expressions first

When solving linear equations the goal is to determine what value, if any, will solve the equation. A general guideline is to use the order of operations to simplify the expressions on both sides first.

Solve for x: 5 ( 3 x + 2 ) 2 = 2 ( 1 7 x ) 5 ( 3 x + 2 ) 2 = 2 ( 1 7 x ) Distribute . 15 x + 10 2 = 2 + 14 x Add same side like terms. 15 x + 8 = 2 + 14 x 15 x + 8 14 x = 2 + 14 x 14 x Subtract 14x on both sides. x + 8 = 2 x + 8 8 = 2 8 Subtract 8 on both sides. x = 10 The solution set is { 10 } .

Video Example 02

Conditional equations, identities, and contradictions

There are three different kinds of equations defined as follows.

Conditional Equation
A conditional equation is true for particular values of the variable.
Identity
An identity is an equation that is true for all possible values of the variable. For example, x = x has a solution set consisting of all real numbers, .
Contradiction
A contradiction is an equation that is never true and thus has no solutions. For example, x + 1 = x has no solution. No solution can be expressed as the empty set {    } = .

So far we have seen only conditional linear equations which had one value in the solution set. If when solving an equation and the end result is an identity, like say 0 = 0, then any value will solve the equation. If when solving an equation the end result is a contradiction, like say 0 = 1, then there is no solution.

Solve for x: 4 ( x + 5 ) + 6 = 2 ( 2 x + 3 ) 4 ( x + 5 ) + 6 = 2 ( 2 x + 3 ) Distribute 4 x + 20 + 6 = 4 x + 6 Add same side like terms . 4 x + 26 = 4 x + 6 4 x + 26 4 x = 4 x + 6 4 x Subtract 4x on both sides. 26 = 6 False There is no solution, .

Solve for y: 3 ( 3 y + 5 ) + 5 = 10 ( y + 2 ) y 3 ( 3 y + 5 ) + 5 = 10 ( y + 2 ) y Distribute 9 y + 15 + 5 = 10 y + 20 y Add same side like terms . 9 y + 20 = 9 y + 20 9 y + 20 20 = 9 y + 20 20 Subtract 20 on both sides . 9 y = 9 y 9 y 9 y = 9 y 9 y Subtract 9y on both sides . 0 = 0 True The equation is an identity, the solution set consists of all real numbers, .

Linear literal equations

Literal equations, or formulas, usually have more than one variable. Since the letters are placeholders for values, the steps for solving them are the same. Use the properties of equality to isolate the indicated variable.

Solve for a: P = 2 a + b P = 2 a + b P b = 2 a + b b Subtract b on both sides. P b = 2 a P b 2 = 2 a 2 Divide both sides by 2. P b 2 = a Solution: a = P b 2

Solve for x: z = x + y 2 z = x + y 2 2 z = 2 x + y 2 Multiply both sides by 2 . 2 z = x + y 2 z y = x + y y Subtract y on both sides . 2 z y = x Solution x = 2 z y

Exercises

Checking solutions

Is  x = 7  a solution to 3 x + 5 = 16 ?

Yes

Is  x = 2  a solution to  2 x 7 = 28 ?

No

Is  x = 3  a solution to  1 3 x 4 = 5 ?

Yes

Is  x = 2  a solution to  3 x 5 = 2 x 15 ?

Yes

Is  x = 1 2  a solution to  3 ( 2 x + 1 ) = 4 x 3 ?

No

Solving in one step

Solve for x:   x 5 = 8

x = 3

Solve for y:   4 + y = 9

y = 5

Solve for x:   x 1 2 = 1 3

x = 5 6

Solve for x:   x + 2 1 2 = 3 1 3

x = 5 6

Solve for x:   4 x = 44

x = 11

Solve for a:  3 a = 30

a = 10

Solve for y:   27 = 9 y

y = 3

Solve for x:   x 3 = 1 2

x = 3 2

Solve for t:   t 12 = 1 4

t = 3

Solve for x:   7 3 x = 1 2

x = 3 14

Solve in two steps

Solve for a:   3 a 7 = 23

a = 10

Solve for y:   3 y + 2 = 13

y = 5

Solve for x:   5 x + 8 = 8

x = 0

Solve for x:   1 2 x + 1 3 = 2 5

x = 2 15

Solve for y:   3 2 y = 11

y = 7

Solve for x:   10 = 2 x 5

x = 5 2

Solve for a:   4 a 2 3 = 1 6

a = 1 8

Solve for x:   3 5 x 1 2 = 1 10

x = 1

Solve for y:   4 5 y + 1 3 = 1 15

y = 1 3

Solve for x:   x 5 = 2

x = 3

Solve in multiple steps

Solve for x:   3 x 5 = 2 x 17

x = 12

Solve for y:   2 y 7 = 3 y + 13

y = 4

Solve for a:   1 2 a 2 3 = a + 1 5

a = 26 15

Solve for x:   2 + 4 x + 9 = 7 x + 8 2 x

x = 1

Solve for a:   3 a + 5 x = 2 a + 7

No Solution,

Solve for b:   7 b + 3 = 2 5 b + 1 2 b

All Reals,

Solve for y:   5 ( 2 y 3 ) + 2 = 12

y = 1 2

Solve for x:   3 2 ( x + 4 ) = 3 ( 4 x 5 )

x = 2

Solve for a:   3 ( 2 a 3 ) + 2 = 3 ( a + 7 )

a = 10 9

Solve for x:   10 ( 3 x + 5 ) 5 ( 4 x + 2 ) = 2 ( 5 x + 20 )

All Reals,

Literal equations

Solve for w:   P = 2 l + 2 w

w = P 2 l 2

Solve for b:   P = a + b + c

b = P a c

Solve for C:   F = 9 5 C + 32

C = 5 F 160 9

Solve for r:   C = 2 π r

r = C 2 π

Solve for y:   z = x y 5

y = 5 z + x

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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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