Solving Systems of Equations using Substitution

Why do we need to learn another method? To answer that, let’s write and solve a system of equations for the following: Jaden and Emily went to Swaner’s Sandwiches to buy lunch for their friends, but Swaner’s Sandwiches doesn’t have any prices. Jaden purchased 4 sodas and 8 sandwiches which cost $41. Emily purchased 8 sodas and 4 sandwiches for $31. What is the cost of each soda and sandwich? Let x represent the cost of a soda and let y represent the cost of a sandwich.

Jaden Emily Equation in Standard Form Equation in Standard Form 4 sodas and 8 sandwiches for $41 8 sodas and 4 sandwiches for $31 Equation in Standard Form Equation in Standard Form Equation in Slope Intercept Equation in Slope Intercept

Jaden Emily 4x + 8y = 41 8x + 4y = 31 y = - 1 2 x + 5 1 8 4 sodas and 8 sandwiches for $41 8 sodas and 4 sandwiches for $31 Equation in Standard Form Equation in Standard Form 4x + 8y = 41 8x + 4y = 31 Equation in Slope Intercept Equation in Slope Intercept y = - 1 2 x + 5 1 8 y = -2x + 7 3 4

y = - 1 2 x + 5 1 8 y = -2x + 7 7 8

y = - 1 2 x + 5 1 8 y = -2x + 7 3 4

Solving Systems of Equations using Substitution Steps: 1. Solve one equation for one variable (y= ; x= ; a=) 2. Substitute the equation from step one into the other equation. 3. Simplify and solve the equation. 4. Substitute back into either original equation to find the value of the other variable. 5. Check the solution in both equations of the system.

y = 4x and 3x + y = -21 Step 1: Solve one equation for one variable. y = 4x (This equation is already solved for y.) Step 2: Substitute the expression from step one into the other equation. 3x + y = -21 3x + 4x = -21 Step 3: Simplify and solve the equation. 7x = -21 x = -3

y = 4x 3x + y = -21 You found that x = -3 Step 4: Substitute back into either original equation to find the value of the other variable. (if x = -3 then y = ?) 3x + y = -21 3(-3) + y = -21 -9 + y = -21 y = -12 Solution to the system is (-3, -12).

y = 4x 3x + y = -21 3x + y = -21 y = 4x 3(-3) + (-12) = -21 Step 5: Check the solution in both equations. Solution to the system is (-3,-12). 3x + y = -21 3(-3) + (-12) = -21 -9 + (-12) = -21 -21= -21 y = 4x -12 = 4(-3) -12 = -12

x + y = 5 and y = 3 + x

Jaden Emily 4x + 8y = 41 8x + 4y = 31 y = - 1 2 x + 5 1 8 4 sodas and 8 sandwiches for $41 8 sodas and 4 sandwiches for $31 Equation in Standard Form Equation in Standard Form 4x + 8y = 41 8x + 4y = 31 Equation in Slope Intercept Equation in Slope Intercept y = - 1 2 x + 5 1 8 y = -2x + 7 3 4

y = - 1 2 x + 5 1 8 y = -2x + 7 3 4

Systems of Equation by Substitution that Require Manipulation

Example #2: x + y = 10 y = -x +10 5x - y = 2 5x -(-x +10) = 2 x + y = 10 and 5x – y = 2 Step 1: Solve one equation for one variable. x + y = 10 y = -x +10 Step 2: Substitute the expression from step one into the other equation. 5x - y = 2 5x -(-x +10) = 2

5x -(-x + 10) = 2 5x + x -10 = 2 6x -10 = 2 6x = 12 x = 2 x + y = 10 and 5x – y = 2 Step 3: Simplify and solve the equation. 5x -(-x + 10) = 2 5x + x -10 = 2 6x -10 = 2 6x = 12 x = 2

Solution to the system is (2,8). x + y = 10 and 5x – y = 2 You found that x = 2 Step 4: Substitute back into either original equation to find the value of the other variable. x + y = 10 2 + y = 10 y = 8 Solution to the system is (2,8).

5x – y = 2 x + y =10 5(2) - (8) = 2 2 + 8 =10 10 – 8 = 2 10 =10 2 = 2 Step 5: Check the solution in both equations. Solution to the system is (2, 8). 5x – y = 2 5(2) - (8) = 2 10 – 8 = 2 2 = 2 x + y =10 2 + 8 =10 10 =10

2x – 3y = 7 2x – y = 5

Ashley’s school is selling tickets to a spring play Ashley’s school is selling tickets to a spring play. On the first day of ticket sales the school sold 3 adult tickets and 9 child tickets for a total of $75. The school took in $67 on the second day by selling 8 adult tickets and 5 child tickets. What is the price each of one senior citizen ticket and one child ticket? Let x represent the cost of an adult ticket and y represent the cost of a child's ticket

System of Equation Word Problems by Substitution

You purchase 8 gal of paint and 3 brushes for $152. 50 You purchase 8 gal of paint and 3 brushes for $152.50.The next day, you purchase 6 gal of paint and 2 brushes for $113.00. Let x equal the cost of a gallon of paint and let y equal the cost of a brush. How much does each gallon of paint and each brush cost?

Shopping at Savers, Lisa buys her children four shirts and three pairs of pants for $85.50. She returns the next day and buys three shirts and five pairs of pants for $115.00. Let x equal the cost of a shirt, and let y equal the cost of a pair of pants. What is the price of each shirt and each pair of pants?

A Little Bit Different We need two equations. How can we get the other? Kristin spent $131 on shirts. Fancy shirts cost $28 and plain shirts cost $15. Let x represent the number of fancy shirts, and let y represent the number of plain shirts. If she bought a total of 7 then how many of each kind did she buy?

Sally has 20 coins in her piggy bank, all dimes and quarters Sally has 20 coins in her piggy bank, all dimes and quarters. The total amount of money is $3.05. Let x represent the number of dimes, and let y represent the number of quarters. How many of each coin does she have?

A group of people bought movies tickets at the AMC Century City A group of people bought movies tickets at the AMC Century City. They bought a total of 7 tickets, some adult and some kid tickets. They spent a total of $72. If adult tickets cost $12 and kid tickets cost $9, how much of each were purchased?

A Honda dealership sells both motorcycles and cars A Honda dealership sells both motorcycles and cars. There are a total of 200 vehicles on the dealership’s lot. The detailer cleaned all the wheels of all the vehicles, which totaled 698 wheels. How many motorcycles are there on the lot?