3.1 linear equations: one transformation linear equations in one variable a linear equation in one...

3.1 Linear Equations: One TransformationLinear Equations in One Variable

A linear equation in one variable is an equation that can be written in the form

ax + by = c, where a, b, and c are real numbers and a ≠ 0.An equation is a statement with an equals sign.

3.1 Linear Equations:One Transformation

Original

Equation

Equivalent

Equation

1. Add the same number to both sides. x - 4 = 7 x = 11

2. Subtract the same number to both sides. x + 2 = 9 x = 7

3. Multiply both sides by the same number. x/2 = 4 x = 8

4. Divide both sides by the same number. 5x = 20 x = 4

5. Interchange the two sides. x = 7 7 = x

3.1 Linear Equations:One Transformation Two equations are equivalent if they have the

same solution set. In solving an equation, the goal is to isolate the

variable by using inverse operations.

3.1 Linear Equations:One Transformation

Solve x + 3 = 7 x + 3 - 3 = 7 - 3 Add -3 to both sides x = 4 Simplify

3.1 Linear Equations:One Transformation

Solve 3x = -12

3x

3

12

3 Divide both sides by 3

x 4 Simplify

3.1 Linear Equations:One Transformation

Solve -4 = x - 7

4 + 7 = x - 7 + 7 Add 7 to both sides.

3 x Simplify

x 3 Transpose

3.1 Linear Equations:One TransformationSolve

x

5= -9

x

5

5 = -9(5) Multiply both sides by 5.

x 45 Simplify

3.1 Linear Equations:One Transformation

Properties of Equalities

1. If a = b, then a + c, = b + c Addition Property

2. If a = b, then a - c, = b - c Subtraction Property

3. If a = b, then ac = bc Multiplication Property

4. If a b and c 0, then a

c

b

c Division Property

3.2 Linear Equations: Two or More Transformations Using Two or More Transformations

1. Simplify both sides of the equation.

2. Use inverse operations to isolate the variable.

3. Check the solution.

3.2 Linear Equations:Two or More TransformationsSolve

4x + 7 - 2x =13

2x 7 13 Combine like terms.

2x 6 Subtract 7 from both sides.

x 3 Divide both sides by 2.

3.2 Linear Equations:Two or More TransformationsSolve

5x +2(3 - x) =15

5x 6 2x 15 Remove parentheses

3x 6 15 Combine like terms.

3x 9 Subtract 6 from both sides.

x 3 Divide both sides by 3.

3.2 Linear Equations: Two or More Transformations

Solve: 3x + 6 = 12

3x + 6 - 6 = 12 – 6 Add -6 to both sides

3x = 6 Combine like terms

x = 2 Divide both sides by 3

3.2 Linear Equations: Two or More Transformations

Solve: 4.5 = 3 + 2x

4.5 - 3 = 3 - 3 + 2x Add -3 to both sides

1.5 = 2x Divide both sides by 2

.75 = x

3.2 Linear Equations: Two or More Transformations

Solve: -2x + 2 - 4x = 20

-6x + 2 = 20 Combine like terms

-6x + 2 – 2 = 20 – 2 Add -2 to both sides

-6x = 18 Divide by -6

x = -3

3.2 Linear Equations: Two or More Transformations

Solve: 4(x - 2) = -10

4x - 8 = -10 Remove parenthesis

4x – 8 + 8 = -10 + 8 Add 8 to both sides

4x = -2 Combine like terms

x = -2/4 = -1/2 Divide both sides by -4

3.2 Linear Equations: Two or More Transformations

Solve:

-3(x - 2) = 21

-3x + 6 = 21 Remove parenthesis

-3x + 6 - 6 = -21 - 6 Add -6 to both sides

-3x = -27 Combine like terms

x = 9 Divide both sides by -3

3.3 Solving Equations with Variables on Both Sides

Collect variables on the side with the greatest variable coefficient

3x 5 8x 30

3x 5 3x 8x 30 3x

5 5x 30

35 5x

7 x

3x 5 8x 30

3x 5 8x 8x 30 8x

5x 5 30

5x 35

x 7

3.3 Solving Equations with Variables on Both Sides

Solve:

4(y 2) 6y 2 8y

4y 8 2y 2

4y 8 2y 2y 2 2y

6y 8 2

6y 8 8 2 8

6y 6

y 1

3.3 Solving Equations with Variables on Both Sides

Solve:

3(x 5) 2x 10 4x

3x 15 2x 10

5x 15 10

5x 5

x 1

3.4 Problem SolvingGeneral Strategies for Problem Solving

1. Read and understand the problem.Choose a variableConstruct a diagram

2. Translate the problem into an equation. 3. Solve the equation. 4. Interpret the results.(Check your answer)

3.5 Solving Equations That Involve Decimals

12.3x - 5.1 = 17

Solve:

12.3x - 5.1+5.1 = 17 + 5.1

12.3x = 22.1

x 22.1

12.3

x 1.796747

3.6 Literal Equations

Solve for v d =vt

Solve for l 2w +2l =P

2l =P - 2w

Solve for x y = mx + b

l P 2w

2

y b mx

y b

mx

d

tv

3.7 Scatter Plots

Quadrant IQuadrant II

Quadrant III Quadrant IV

3.7 Scatter Plots

A(3,2)

BB(-1,-3)A

C(-5,0)B