Unit 2Reasoning with Equations and Inequalities

Warm-UpSolve the following equations:

Graph Using Slope-Intercept Form

Steps

1. Solve for slope-intercept form:

2. Graph the y-intercept

3. Use the slope to find the next point

4. Use the points to draw a line!

Slope

slope ¿riserun

Positive Slope = UP to the right Negative Slope = DOWN to the

right

Graph

• What is your slope?

• What is your y-intercept?

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

System of 2 Linear Equations

• 2 equations with 2 variables (x & y) each

Ax + By = C Dx + Ey = F

A, B, C, D, E, and F are all numbers

• Solution of a System – an ordered pair, (x,y) that makes both equations true

Check whether the ordered pairs are solutions of the system:

x - 3y = -5-2x + 3y = 10

a) (1,4)1-3(4)= -51-12= -5-11 = -5*doesn’t work in the 1st equation, no need to check the 2nd

Not a solution

b)(-5,0)-5-3(0)= -5-5 = -5

-2(-5)+3(0)=10

10=10

Solution

Solving Systems by Graphing

1. Graph each equation on the same coordinate plane.

2. If the lines intersect: The point (ordered pair) where the lines intersect is the solution.

Solve the system graphically:y = 3x – 12y = -2x + 3

How do you check the solution?

Solving Systems by Graphing

Types of solutions:• If the lines have the same y-intercept ,

and the same slope , then the system is dependent. The lines fall on top of each other!

• If the lines have the same slope , but different y-intercepts , then system is inconsistent. Parallel lines have no solutions!

• If the lines have different slopes , the system is independent.

Independent Inconsistent Dependent

Solving Systems by Elimination

1. Arrange the equations with like terms in columns.

2. Multiply, if necessary, to create opposite coefficients for one variable.

3. Add/Subtract the equations.

4. Substitute the value to solve for the other variable.

5. Write your answer as an ordered pair.

6. Check your answer.

2x – 2y = -82x + 2y = 4

4x = -4 4 4 x = -1

2(-1) – 2y = -8 -2– 2y = -8

+2 +2 – 2y = -6

-2 -2y = 3

The solution is (-1,3).

Solving Systems by Substitution

1. Solve an equation for one variable Isolate or

2. Substitute Plug what or is into the other equation

3. Solve the equation Reverse PEMDAS

4. Plug back to find the other variable Plug the value into the equation and solve

5. Check your solution Plug the solution back into both equations

2x – 3y = -2y = -x + 4 is already by itself!

2x – 3(-x + 4) = -22x + 3x - 12 = -2

5x – 12 = -25x = 10

x = 2

y = -2 + 4

y = 2

The solution is (2,2).

Word Problems

1. Define variables

2. Write as a system of equations.

3. Solve showing all steps.

4. State your solutions in words.

Example 1: Work Schedule

You worked 18 hours last week and earned a total of $124 before taxes. Your job as a lifeguard pays $8 per hour, and your job as a cashier pays $6 per hour. How many hours did you work at each job?

x + y = 18

8x + 6y = 124

x = hours as lifeguard

y = hours as cashier

Graphing a Linear Inequality

1. Sketch the line given by the equation (solid if ≥ or ≤, dashed if < or >). This line separates the coordinate plane into 2 halves. – In one half-plane – all of the points

are solutions of the inequality. – In the other half-plane - no point is

a solution2. You can decide which half to shade

by testing ONE point.3. Shade the half that has the

solutions to the inequality.

𝑦 ≥ (−𝟑𝟐 )𝒙+1

Graphing Systems of Linear Inequalities

1. Graph the lines and appropriate shading for each inequality on the same coordinate plane.

2. Remember: dotted for < and >, and solid for and .

3. The solution is the section where all the shadings overlap.

5

2 4 4

x y

x y

Ticket Out the Door

Solve the following equations:

1.

2.

3.