Chapter 3 – Systems of Linear Equations

Using a Graph to Solve Linear Systems of Equations One Day

What is a System of Linear Equations? A system of linear equations is simply two or more linear equations using the same variables. If the system of linear equations has a solution, then the solution will be an ordered pair (x, y) where x and y make both equations true at the same time. We will be dealing with systems of 2 equations with 2 variables as well as systems that have 3 equations and 3 variables (x, y, z).

Cases for System of Equations

Solving Systems by Graphing Consider the following system: x y Anytime we solve a system of equations we must check our solution. We will do this by substituting the solution back into each equation for x and y. (1 , 2) x – y = –1 (1) – (2) = –1  x + 2y = 5 (1) + 2(2) = 1 + 4 = 5 

Graphing to Solve a Linear System Solve the following system by graphing: 3x + 6y = 15 –2x + 3y = –3 x y Using the slope intercept form of these equations, we can graph them carefully on graph paper. (3 , 1) Label the solution! Lastly, we need to verify our solution is correct, by substituting (3 , 1).

Steps for Solving by Graphing 1. Put each equation into y=mx+b (solve for y) 2. Graph each equation on graph paper. Precision is important, use a ruler!! 3. Determine the point of intersection, estimate if necessary. 4. Check your solution in both equations!

Your turn… Solve the following system of equations by graphing.

Your turn… Solve the following system of equations by graphing.

Homework pg 120 (# 1-9 odd, 25-35 odd, 44)

Solving by Substitution and Elimination Four Days

Solving Linear Systems There are two methods of solving a system of equations algebraically: Elimination Substitution

Steps for Solving by Substitution To solve a system of equations by substitution… 1. Solve one equation for one of the variables. 2. Substitute the value of the variable into the other equation. 3. Simplify and solve the equation for the remaining variable. 4. Substitute back into either equation to find the value of the other variable.

Substitution Solve the system: x - 2y = -5 y = x + 2 Notice: One equation is already solved for one variable. Substitute (x + 2) for y in the first equation. x - 2y = -5 x - 2(x + 2) = -5 We now have one equation with one variable. Simplify and solve. x - 2x – 4 = -5 -x - 4 = -5 -x = -1 x = 1 Substitute 1 for x in either equation to find y. y = x + 2 y = 1 + 2 so y = 3 The solution is (1, 3)

Substitution Let’s check the solution. The answer (1, 3) must check in both equations. x - 2y = -5 y = x + 2 1 - 2(3) = -5 3 = 1 + 2 -5 = -5 3 = 3 

Substitution Solve the systems by substitution: 1. x = 4 2x - 3y = -19

Steps for Solving by Elimination 1. Write each equation in standard form Ax+By=C. 2. Determine which variable you want to eliminate. 3. Multiply an entire equation by a value that will result in the terms you want to eliminate being additive inverses. 4. Add the equations. The result is one equation with one variable. 5. Solve the resulting equation. 6. Substitute the solution into one of the original equations and solve for the remaining variable.

Elimination Solve the system: 3s - 2t = 10 4s + t = 6 We could multiply the second equation by 2 and the t terms would be inverses. OR We could multiply the first equation by 4 and the second equation by -3 to make the s terms inverses. Let’s multiply the second equation by 2 to eliminate t. (It’s easier.) 3s - 2t = 10 3s – 2t = 10 2(4s + t = 6) 8s + 2t = 12 Add and solve: 11s + 0t = 22 11s = 22 s = 2 Insert the value of s to find the value of t 3(2) - 2t = 10 t = -2 The solution is (2, -2).

Elimination Solve the system by elimination: 1. -4x + y = -12 4. 5m + 2n = -8 4m +3n = 2

Homework pg 128 (# 1-7 odd, 19-27 odd, 31-37 odd)

Applications of Systems of Equations The sum of two numbers is 70 and their difference is 24. Find the two numbers.

Applications of Systems of Equations Two angles are supplementary. The measure of one angle is 10 degrees more than three times the other. Find the measure of each angle.

Applications of Systems of Equations Two groups go on a whitewater rafting trip. Group 1 rented 6 rafts and 8 kayaks for a total of $510. Group 2 rented 3 rafts and 11 kayaks for a total of $465. How much did it cost to rent each raft and each kayak?

Homework Applications #1

Homework Applications #2

Homework Applications #3

Systems of Linear Inequalities Two Days

Warm-up Complete the following warm-up while I check homework: Graph the following inequalities:

Solving Systems of Linear Inequalities 1. We show the solution to a system of linear inequalities by graphing them and determining the region that satisfies all of the individual inequalities simultaneously. a. This process is easier if we put the inequalities into Slope-Intercept Form, y = mx + b 2. Graph the line using the y-intercept & slope. a. If the inequality is < or >, make the lines dotted. b. If the inequality is < or >, make the lines solid. 3. The solution also includes points not on the line, so you need to shade the region of the graph: a. above the line for ‘y >’ or ‘y ’. b. below the line for ‘y <’ or ‘y ≤’.

Example Solve and graph the following system of inequalities.

Solving Systems of Linear Inequalities a: 3x + 4y > - 4

Solving Systems of Linear Inequalities a: 3x + 4y > - 4 b: x + 2y < 2

Solving Systems of Linear Inequalities The region that satisfies both equations is the area of overlap. This is the solution to our system of inequalities. Any point in this region satisfies the system. a: 3x + 4y > - 4 b: x + 2y < 2

Solving Systems of Linear Inequalities Lets solve the following:

Applications Jane’s band wants to spend no more than $575 recording their CD. The studio charges at least $35 per hour to record. Write and graph a system of inequalities to represent this situation.

Jane’s Band…

Applications The most Dave can spend on hot dogs and bun for a cookout is $42. A package of 10 hot dogs costs $3.50. A package of 8 buns costs $2.50. He needs to buy at least 40 hot dogs and 40 buns. Write and graph a system of inequalities that describes this situation. Give 3 examples of different purchases he can make and still satisfy the requirements.

Dave’s Cookout

Your Turn Graph the following system of linear inequalities:

Your Turn Graph the following system of linear inequalities:

Your Turn Graph the following system of linear inequalities:

Your Turn You work two jobs and can work no more than a total of 25 hours per week. You make $8/hr at the first job and $10/hr at the second job. Your boss at the second job can only give you 10 hours each week. Write and graph a system of inequalities assuming that you need to earn at least $150/week.

Two Jobs Problem

Homework pg 136 (# 1-17 odd)

Homework Practice 3-3 WS (# 1-13 odd)

Systems of Three Equations and Three Variables One Day

Solving Systems of Three Equations Algebraically When we have three equations in a system, we can use the same two methods to solve them algebraically as with two equations. Whether you use substitution or elimination, you should begin by numbering the equations!

Solving Systems of Three Equations Linear Combination Method Choose two of the equations and eliminate one variable as before. Now choose one of the equations from step 1 and the other equation you didn’t use and eliminate the same variable. You should now have two equations (one from step 1 and one from step 2) that you can solve by elimination. Find the third variable by substituting the two known values into any equation.

Example

Example

Example

Homework pg 157 (# 1-9 odd, 10)

Solving Systems of Three Equations Substitution Method Choose one of the three equations and isolate one of the variables. Substitute the new expression into each of the other two equations. These two equations now have the same two variables. Solve this 2 x 2 system as before. Find the third variable by substituting the two known values into any equation.