Reasoning with Equations and Inequalities Unit 2 Reasoning with Equations and Inequalities

Solve the following equations: 𝟔𝒙−𝟕=𝒙+𝟑 −𝟑𝒙=𝟏𝟓 𝟐𝒙 𝟑 =𝟒 𝟐𝒙−𝟒=𝟏𝟎 Warm-Up Solve the following equations: 𝟔𝒙−𝟕=𝒙+𝟑 −𝟑𝒙=𝟏𝟓 𝟐𝒙 𝟑 =𝟒 𝟐𝒙−𝟒=𝟏𝟎

Graph Using Slope-Intercept Form Steps Solve for slope-intercept form: 𝒚= 𝒎𝒙+𝒃 Graph the y-intercept 𝒃 Use the slope 𝒎 to find the next point Use the points to draw a line!

Slope = rise run slope =𝑚 Positive Slope = UP to the right Negative Slope = DOWN to the right

Graph 𝑦=2𝑥+3 What is your slope? What is your y- intercept?

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 (-5,0) -5-3(0)= -5 -5 = -5 -2(-5)+3(0)=10 10=10 Solution (1,4) 1-3(4)= -5 1-12= -5 -11 = -5 *doesn’t work in the 1st equation, no need to check the 2nd Not a solution

Solving Systems by Graphing Graph each equation on the same coordinate plane. If the lines intersect: The point (ordered pair) where the lines intersect is the solution.

Solving Systems by Graphing Make sure each equation is in slope- intercept form: 𝑦=𝑚𝑥+𝑏 Graph each equation on the same graph paper. The point where the lines intersect is the solution. If they don’t intersect then there’s no solution. Check your solution algebraically!

How do you check the solution? Solve the system graphically: y = 3x – 12 y = -2x + 3 (3, -3) How do you check the solution?

Solving Systems by Graphing Types of solutions: If the lines have the same y-intercept 𝑏, and the same slope 𝑚, then the system is dependent. The lines fall on top of each other! If the lines have the same slope 𝑚, but different y-intercepts 𝑏, then system is inconsistent. Parallel lines have no solutions! If the lines have different slopes 𝑚, the system is independent.

Independent Inconsistent Dependent

Solving Systems by Elimination Arrange the equations with like terms in columns. Multiply, if necessary, to create opposite coefficients for one variable. Add/Subtract the equations. Substitute the value to solve for the other variable. Write your answer as an ordered pair. Check your answer.

2x – 2y = -8 2x + 2y = 4 2(-1) – 2y = -8 4x = -4 -2– 2y = -8 4 4 +2 +2 x = -1 – 2y = -6 -2 -2 y = 3 The solution is (-1,3).

Solving Systems by Substitution Solve an equation for one variable Isolate 𝑥 or 𝑦 Substitute Plug what 𝑥 or 𝑦 is into the other equation Solve the equation Reverse PEMDAS Plug back to find the other variable Plug the value into the equation and solve Check your solution Plug the solution back into both equations

2x – 3y = -2 y = -x + 4 𝑦 is already by itself! 2x – 3(-x + 4) = -2 2x + 3x - 12 = -2 5x – 12 = -2 5x = 10 x = 2 y = -2 + 4 y = 2 The solution is (2,2).

Word Problems Define variables Write as a system of equations. Solve showing all steps. 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 = hours as lifeguard x + y = 18 8x + 6y = 124 y = hours as cashier

Graphing a Linear Inequality 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 solution 2. You can decide which half to shade by testing ONE point. 3. Shade the half that has the solutions to the inequality.

𝑦 ≥ − 𝟑 𝟐 𝒙+ 1

Graphing Systems of Linear Inequalities Graph the lines and appropriate shading for each inequality on the same coordinate plane. Remember: dotted for < and >, and solid for ≥ and ≤. The solution is the section where all the shadings overlap.

Solve the following equations: Ticket Out the Door Solve the following equations: 1. 3 𝑥−2 =13 7𝑥−9=3𝑥+11 8−2𝑥=12