7.2-3 Solving Linear Equations

A linear equation in one variable is an equation in which the same letter is used in all variable terms and the

exponent of the variable is 1.

The solution to an equation in one variable is the number that

can be substituted in place of the variable and makes the equation

true. For example 5 is a solution to

the equation 2x + 3 = 13 because 2(5) + 3 = 13 is true.

Equivalent equations are equations that have the same

solutions. For example 2x + 3 = 13

and x = 5 are equivalent equations because each has the

solution of 5.

Basic Principle of Equality

• To preserve equality, if an operation is performed on one side of an equation, the same operation must be performed on the other side.

There are 2 principles (axioms) we will use to solve linear equations in

one variable. The first is the addition principle of equality. This

principle allows us to add (or subtract) the same value to both sides of an equation to obtain an

equivalent equation.

To solve an equation using the addition axiom:

• Locate the variable in the equation.

• Identify the constant that is associated with the variable by addition or subtraction.

• Add the opposite of the constant to both sides of the equation.

Solve each equation.

19 x 113 y 38 x 56 y3.22.7 y

Often we need to combine like terms on one or both sides of the

equation before solving.For example:

63 xgives termslike combining

695437 xxx

Whenever variable terms appear on both sides of an equation we use the

addition principle to move all variable terms to the same side, then solve.

For example to solve

8x

xx 685 get tosidesboth 6x to add

When solving these type equations, it makes no difference the side from

which you remove the variable term to start. The goal is to get all variable

terms on one side and all constants on the other.

Solve each equation.

xx 475 567 xx5283 cc

Whenever quantities appear in parentheses on either side of the

equation they must be removed first.

124 x

1283 xx12243 xx

Solve each equation.

1283 xx 917836 xx 44327 uu

The Multiplication Principle of Equality

For each problem so far the coefficient for the variable ended

up being one. We use the multiplication principle (axiom)

to solve equations where the coefficient of the variable is not

one.

The multiplication principle allows us to multiply (or divide) each side of an equation by the same nonzero

quantity to obtain an equivalent equation.

The goal is to get +1 times the variable = a number.

To solve an equation where the coefficient of the variable is not one you need to multiply both

sides of the equation by the reciprocal of the coefficient. An alternate way is to divide both sides by the coefficient of the

variable.

Solve each equation.

248 x 255 x x642 w1872 411 q 03 x 16

9

4x 15

7

3 x

5

2

4

3y 8.168.2 m

Sometimes it is necessary to combine like terms before

solving the equation.

4538 xx

455 x termslike combine

5by sidesboth divide

9x

Solve each equation.

48529 www 676652 sss

When more than one operation is indicated on the variable, undo

addition or subtraction first, then undo multiplication or division next.

Solve: 4x – 2 = 18

• Since the variable has been multiplied by 4 and subtracted by 2, undo by adding 2 and dividing by 4.

4 2 18

4 20

5

4 5 2 18

18

4

1

2

4

8

2x

x

x

Solve: 118 – 22m = 30• Think of 118 – 22m as

118 + ( - 22m)

118 22 30

22 88

22

118 118

2

88

4

118 22 4 30

118 88 30

3

2

30

2

0

2

m

m

m

m

Solve: 5x – 4 = 8x – 13 5 4 5 8 13 5

4 3 13

4 13 3 13 13

9 3

9 3

3 33

x x x x

x

x

x

x

x

5 4 8 8 13 8

3 4 13

3 4 4 13 4

3 9

3 9

3 33

x x x x

x

x

x

x

x

Summary of steps for solving an equation:

• Remove parentheses.• Combine like terms on each side of the

equation.• Sort terms to collect the variable terms on

one side and constants on the other.• Solve for the variable by multiplying by the

reciprocal of the coefficient or dividing by the coefficient of the variable.

5 ( h – 4 ) + 2 = 3h – 4

5h – 20 + 2 = 3h – 4 Distribute.

5h – 18 = 3h – 4

Step 1

Step 2

Combine terms.

5h – 18 + 18 = 3h – 4 + 18 Add 18.

5h = 3h + 14 Combine terms.

Subtract 3h.5h – 3h = 3h + 14 – 3h

2 2

Combine terms.2h = 14

=

h = 7

Step 3 2h 14 Divide by 2.

Solve the following equation.

Check by substituting 7 for h in the original equation.Step 4

5 ( h – 4 ) + 2 = 3h – 4

5 ( 7 – 4 ) + 2 = 3(7) – 4

5 (3) + 2 = 3(7) – 4

15 + 2 = 21 – 4

17 = 17

? Let h = 7.

? Subtract.

True

? Multiply.

The solution to the equation is 7.

Solving an Equation That Has Infinitely Many Solutions

4 ( 2n + 6 ) = 2 ( 3n + 12 ) + 2n

8n + 24 = 6n + 24 + 2n Distribute.

8n + 24 = 8n + 24 Combine terms.

8n + 24 – 24 = 8n + 24 – 24 Subtract 24.

8n = 8n Combine terms.

Subtract 8n.8n – 8n = 8n – 8n

True0 = 0

An equation with both sides exactly the same, like 0 = 0, is called an identity. An identity is true for all replacements of the variables. We

indicate this by writing all real numbers.

Solving an Equation That Has No Solution

6x – ( 4 – 3x ) = 8 + 3 ( 3x – 9 )

6x – 4 + 3x = 8 + 9x – 27 Distribute.

9x – 4 = –19 + 9x Combine terms.

9x – 4 – 9x = –19 + 9x – 9x Subtract 9x.

– 4 = –19 False

Again, the variable has disappeared, but this time a false statement (– 4 = – 19) results. Whenever this happens in solving an

equation, it is a signal that the equation has no solution and we write no solution.

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