1.3 Solving Linear Equations1.3 Solving Linear Equations

What is an equation?What is an equation?• A statement in which 2 expressions are =A statement in which 2 expressions are =

ExEx: Which of the following are equations?: Which of the following are equations?

a.a. 3x-7=123x-7=12 b. 24x+5b. 24x+5

c.c. 2x-7x2x-7x22+4x+4x33 d. 12x+3= -4x-8d. 12x+3= -4x-8

Properties of EqualityProperties of Equality• Addition prop of =Addition prop of = - can add the same - can add the same

term to both sides of an equation.term to both sides of an equation.

• Subtraction prop of =Subtraction prop of = - can subtract the - can subtract the same term from both sides of an equation.same term from both sides of an equation.

• Multiplication prop of =Multiplication prop of = - can multiply both - can multiply both sides of an equation by the same term.sides of an equation by the same term.

• Division prop of =Division prop of = - can divide both sides - can divide both sides of an equation by the same term.of an equation by the same term.

** So basically, whatever you do to one side ** So basically, whatever you do to one side of an equation, you MUST do to the other!of an equation, you MUST do to the other!

To solve an equation for a variable:To solve an equation for a variable:

• Do order of operations backwards (undo Do order of operations backwards (undo +/- first, then mult/div.)+/- first, then mult/div.)

• Keep going until the variable is by itself on Keep going until the variable is by itself on one side of the equationone side of the equation

• You may have to simplify each side first.You may have to simplify each side first.

Example: Solve for the variable.Example: Solve for the variable.

1689

2x

89

2x

2

98x

36x

xxx 72425

xxx 288105

287105 xx

281012 x

1812 x

12

18x

2

3x

Ex: Solve for x.Ex: Solve for x.

10

32

5

1

3

2 xx

10

3230

5

1

3

230 xx

960620 xx

9640 x

1540 x

40

15

x

8

3x

Ex: Solve the equations.Ex: Solve the equations.5(x-4)=5x+125(x-4)=5x+12

5x-20=5x+125x-20=5x+12

-20=12-20=12

Doesn’t make sense!Doesn’t make sense!

Answer: No solutionAnswer: No solution

7x+14 -3x=4x+147x+14 -3x=4x+14

4x+14=4x+144x+14=4x+14

0=00=0

This one makes sense, This one makes sense, but there’s no variable but there’s no variable left!left!

Answer: All real numbersAnswer: All real numbers

Dry ice is solid CODry ice is solid CO22. It does not melt, but . It does not melt, but

changes into a gas at -109.3changes into a gas at -109.3ooF. What is F. What is this temperature in this temperature in ooC?C?

325

9F CUse

325

93.109 C

C5

93.141

C )3.141(9

5

Co 5.78

ExamplesExamples

• Solve 11x-9y= -4 for y.Solve 11x-9y= -4 for y.

-11x -11x-11x -11x

-9y=-11x-4-9y=-11x-4

-9-9 -9 -9-9 -9

• Solve 7x-3y=8 for x.Solve 7x-3y=8 for x.

+3y +3y+3y +3y

7x=3y+87x=3y+8

7 7 77 7 7

9

4

9

11 xy

7

8

7

3 yx

Ex: Solve the area of a trapezoid Ex: Solve the area of a trapezoid formula for bformula for b11..

A = ½ (bA = ½ (b11+b+b22) h) h

2A = (b2A = (b11+b+b22) h) h

21

2bb

h

A

12

2bb

h

A

Last Example:Last Example:

• You are selling 2 types of hats: baseball hats & You are selling 2 types of hats: baseball hats & visors. Write an equation that represents total visors. Write an equation that represents total revenue.revenue.

Total Revenue

Price of baseball

cap

# of caps sold

Price of visor

# of visors sold

R = p1B + p2V