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Use the properties of equality to solve: 5 x = 35 5 x = 35 5 x 5 = 35 5 Divide both sides by -5 . x = 7 Simplify The solution set is { 7 } .

Two other important properties are:

Symmetric property

If  A = B  then  B = A .

Transitive property

If  A = B  and  B = C  then  A = C .

When solving, we often see 2 = x but that is equivalent to x = 2 .

Isolating the variable

The idea behind solving in algebra is to isolate the variable. If given a linear equation of the form a x + b = c then we can solve it in two steps. First use the equality property of addition or subtraction to isolate the variable term. Next isolate the variable by using the equality property of multiplication or division. The property choice depends on the given operation, we choose to apply the opposite property of the given operation. For example, if given a term plus three we would first choose to subtract three on both sides of the equation. If given two times the variable then we would choose to divide both sides by two.

Solve 2 x + 3 = 13 . 2 x + 3 = 13 2 x + 3 3 = 13 3 Subtract 3 on both sides . 2 x = 10 2 x 2 = 10 2 Divide both sides by 2 . x = 5 The solution set is { 5 } .

Solve 3 x 2 = 9 . 3 x 2 = 9 3 x 2 + 2 = 9 + 2 Add 2 to both sides . 3 x = 11 3 x 3 = 11 3 Divide both sides by -3 . x = 11 3 The solution set is { 11 3 }

Solve x 3 + 1 2 = 2 3 . x 3 + 1 2 = 2 3 x 3 + 1 2 1 2 = 2 3 1 2 Subtract  1 2  on both sides . x 3 = 2 3 ( 2 2 ) 1 2 ( 3 3 ) x 3 = 4 6 3 6 x 3 = 1 6 3 x 3 = 3 1 6 Multiply both sides by 3 . x = 1 2 The solution set is { 1 2 } .

In order to retain the equality, we must perform the same operation on both sides of the equation. To isolate the variable we want to remember to choose the opposite operation not the opposite number. For example, if we have -5x = 20 then we choose to divide both sides by -5, not 5.

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Multiplying by the reciprocal

Recall that when multiplying reciprocals the result is 1, for example, ( 3 5 ) ( 5 3 ) = 15 15 = 1 . We can use this fact when the coefficient of the variable is a fraction.

Solve 4 5 x 5 = 15 . 4 5 x 5 = 15 4 5 x 5 + 5 = 15 + 5 Add 5 on both sides . 4 5 x = 20 5 4 ( 4 5 x ) = 5 4 ( 20 ) Multiply both sides by  5 4 1 x = 5 5 Simplify . x = 25 The solution set is { 25 } .

Combining like terms and simplifying

Linear equations typically will not be given in standard form and thus will require some additional preliminary steps. These additional steps are to first simplify the expressions on each side of the equal sign using the order of operations.

Opposite side like terms

Given a linear equation in the form a x + b = c x + d we must combine like terms on opposite sides of the equal sign. To do this we will use the addition or subtraction property of equality to combine like terms on either side of the equation.

Solve for y: 2 y + 3 = 5 y + 17 2 y + 3 = 5 y + 17 2 y + 3 5 y = 5 y + 17 5 y Subtract 5y on both sides . 7 y + 3 = 17 7 y + 3 3 = 17 3 Subtract 3 on both sides . 7 y = 14 7 y 7 = 14 7 Divide both sides by -7 . y = 2 The solution set is { 2 } .

Same side like terms

We will often encounter linear equations where the expressions on each side of the equal sign could be simplified. If this is the case then it is usually best to simplify each side first. After which we then use the properties of equality to combine opposite side like terms.

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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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