two equations are equivalent if they have the same solutions. solving a linear equation an equation...

Two equations are equivalent if they have the same solutions.

Solving a Linear Equation

An equation is a statement in which two expressions are equal.

A linear equation in one variable is an equation that can be written in the form ax = b where a and b are constants and a 0.

A number is a solution of an equation if the statement is true when the number is substituted for the variable.

For instance, the equations x – 4 = 1 and x = 5 are equivalent because both have the number 5 as their only solution.

The transformations, or changes, on the following slide produce equivalent equations and can be used to solve an equation.

Solving a Linear Equation

TRANSFORMATIONS THAT PRODUCE EQUIVALENT EQUATIONS

ADDITION PROPERTY

OF EQUALITY

SUBTRACTION PROPERTY

OF EQUALITY

MULTIPLICATION PROPERTY

OF EQUALITY

DIVISION PROPERTY

OF EQUALITY

Add the same number to both sides:

If a = b, then a + c = b + c.

Subtract the same number from both sides:

If a = b, then a – c = b – c.

Multiply both sides by the same nonzero

number: If a = b and c 0, then ac = bc.

Divide both sides by the same nonzero

number: If a = b and c 0, then a c = b

c.

Your goal is to isolate the variable on one side of the equation.

Write original equation.

Subtract 9 from each side.

Simplify.

SOLUTION

Solving an Equation with a Variable on One Side

Solve + 9 = 15.37

x

+ 9 = 1537

x

= 6 37

x

Multiply each side by , the reciprocal of . 73

37 = 6 x

73

(6)

= 6 x 14

The solution is 14. Substitute 14 for x.

Solution checks.

+ 9 = 1537

( x )

CHECK

?

15 = 15

14

Example

• Fractions (Multiply by LCD)

3

4

4

52

11

4x F

HGIKJ

You do: (multiply by LCD)

3 2 5

4 3 6x

Solve 5n + 11 = 7n – 9.

5n + 11 = 7n – 9

5n + 11 = 2n – 9

5n + 20 = 2n

10 = n Divide each side by 2.

Add 9 to each side.

Subtract 5n from each side.

Write original equation.

SOLUTION

Solving an Equation with a Variable on Both Sides

The solution is 10. Check this in the original equation.

You do:

6x – 4 = 2x + 10

Solve 4(3x – 5) = –2(–x + 8) – 6x.

4(3x – 5) = –2(–x + 8) – 6x

12x – 20 = 2x –16 – 6x

12x – 20 = – 4x – 16

Divide each side by 16.

Combine like terms.

Distributive property

Write original equation.

SOLUTION

Using the Distributive Property

16x – 20 = –16 Add 4x to each side.

16x = 4 Add 20 to each side.

x =14

The solution is . Check this in the original equation.14

You do:

2(x + 2) = 3(x – 8)

Example• Interesting:

3 8 3 24

5 12 6 12

( )x x

x x x

REAL ESTATE A real estate broker’s base salary is $18,000. She earns a 4%

commission on total sales. How much must she sell to earn $55,000 total?

SOLUTION

Verbal Model

Labels

Total income

Commissionrate

Total income = 55,000

Base salary = 18,000

Commission rate = 0.04

(dollars)

(dollars)

(percent in decimal form)

Basesalary

+

Using Linear Equations in Real Life

= •Totalsales

Total sales = x (dollars)

AlgebraicModel 18,000 + 0.04x55,000 =

REAL ESTATE A real estate broker’s base salary is $18,000. She earns a 4%

commission on total sales. How much must she sell to earn $55,000 total?

SOLUTION

18,000 + 0.04x55,000 =

37,000 = 0.04x

925,000 = x

Write linear equation.

Subtract 18,000 from each

side.

Divide each side by 0.04.

The broker must sell real estate worth a total of $925,000 to earn $55,000.

Writing and Using a Geometric Formula

You do:

A car salesperson’s base salary is $21,000. She earns a 5 % commission on sales. How much must she sell to earn $65,000 total?

Example• The state sales tax in Pennsylvania is 0.06

(or 6%). If your total bill at the music store included $1.32 in tax, how much did the merchandise cost?

You have a 3 inch by 5 inch photo that you want to enlarge, mat, and frame.

You want the width of the mat to be 2 inches on all sides. You want the

perimeter of the framed photo to be 44 inches. By what percent should you

enlarge the photo?

SOLUTION

Verbal Model

Labels

Perimeter +

Perimeter = 44 (inches)

(inches)

(inches)

2 • Width= 2 • Length

Width = 4 + 3x

Length = 4 + 5x

44 = 2(4 + 3x) + 2(4 + 5x)Algebraic

Model

Writing and Using a Geometric Formula

44 = 2(4 + 3x) + 2(4 + 5x) Write linear equation.

Distribute and combine like terms.44 = 16 + 16x

Subtract 16 from each side.28 = 16x

Divide each side by

16.

1.75 = x

You have a 3 inch by 5 inch photo that you want to enlarge, mat, and frame.

You want the width of the mat to be 2 inches on all sides. You want the

perimeter of the framed photo to be 44 inches. By what percent should you

enlarge the photo?

SOLUTION

You should enlarge the photo to 175% of its original size.

Writing and Using a Geometric Formula