Solving Linear Equations1.5 through 1.7

• Is 2 + 4 = 6 a true equation?

• Is 3 – 5 = 10 a true equation?

• Is 2x + 1 = 7 a true equation?

Equations

This equation is a conditional equation, since it depends on x.

1. If there are fractions, multiply everything by the LCD.2. Get rid of ( )’s and combine like terms.3. Isolate the variable (get it by itself on one side of

equation).4. Check your answer!

1. If there are fractions, multiply everything by the LCD.2. Get rid of ( )’s and combine like terms.3. Isolate the variable (get it by itself on one side of

equation).4. Check your answer!

Linear Equations

The graph of any linear equation is a straight line.

Solving Linear Equations:

Determine whether 3 is a solution of 2x + 4 = 10

Solution To find out, check the proposed solution, x = 3:

Example 1

2x + 4 = 10 This is the original equation.

2(3) + 4 = 10 Substitute 3 for x.

6 + 4 = 10 Multiply.

10 = 10 Add.

Since 10 = 10, 3 is a solution, or root, of the equation.

Solve 3(x - 2) = 20.

Solution:

Example 3

3(x - 2) = 20 Original equation.

3x – 6 = 20 Use the distributive property.

3x = 26 Add 6 to both sides.

x = Divide both sides by 3.3

26

Solve

Solution:

Example 5

This is the given equation.

Multiply both sides by 6 (LCD).

Use distributive property.

10(x - 3) = 9(x - 2) + 12

2)2()3( 23

35 xx

2)2()3( 23

35 xx

2)2(6)3(6 23

35 xx

26)2(6)3(6 23

35 xx

10x - 30 = 9x - 18 + 12 Use distributive property again.

10x – 30 = 9x - 6 Combine like terms.

x = 24 Isolate variable.

1. If there are fractions, multiply everything by the LCD.2. Get rid of ( )’s and combine like terms.3. Isolate the variable (get it by itself on one side of

equation).4. Check your answer!

1. If there are fractions, multiply everything by the LCD.2. Get rid of ( )’s and combine like terms.3. Isolate the variable (get it by itself on one side of

equation).4. Check your answer!

Linear Equations

The graph of any linear equation is a straight line.

Solving Linear Equations:

Conditional: An equation that is true for at least one value of “x” (it’s sometimes true).

Identity: An equation that is true for all values of “x” (it’s always true).

Contradiction: An equation that has no solution (it’s never true).

Conditional: An equation that is true for at least one value of “x” (it’s sometimes true).

Identity: An equation that is true for all values of “x” (it’s always true).

Contradiction: An equation that has no solution (it’s never true).

Types of Equations

Solve 2(x +1) – x = 3(1 + x) – (2x + 1):

Example 7

2(x +1) – x = 3(1 + x) – (2x + 1) Original equation.

2x + 2 – x = 3 + 3x – 2x – 1 Use the distributive property.

x + 2 = x + 2 Combine like terms.

Last equation is always true for any value of x

it is an identity!

Solve

Solution:

Example 8

Original equation.

Multiply everything by 6 (LCD).

Simplify fractions.

3213

23

31 4 xx x

3213

23

31 4 xx x

3213

23

31 6)(6)4(66 xx x

)213(2)3(3)4(6)1(2 xxx

26x – 2 = 26x + 5 Combine like terms.

-2 = 5 Subtract 26x from both sides.

42692422 xxx Use distributive property.

Since –2 = 5 is never true, there is no solution it is a contradiction!

Determine whether the equation 3(x - 1) = 2(x + 3) + x is an identity, a conditional equation, or a contradiction.

Solution To find out, solve the equation.

Time Trial

3(x – 1) = 2(x + 3) + x

3x – 3 = 2x + 6 + x

3x – 3 = 3x + 6

-3 ≠ 6

This equation is a contradiction.

Formulas

A formula is an equation involving two or more variables.

D = RT

A = LW

P = 2L + 2W

Solve for t in the following formula: A = p + prt.

Solution: To solve for t means to isolate it on on side.

Example 10

A = p + prt Original equation.

A – p = prt To isolate t, subtract p from both sides.

Divide both sides by pr.

Write t on left-hand side.

tprpA

prpAt