Solving Systems of Equations

Solving Systems of Equations

Methods We Have Used to Solve Systems of Equations

• Graphing

• Elimination

• Substitution

Graphing Method

• Use the Desmos Graphing Calculator

• One Solution: Intersection of the two lines

• No Solution: Parallel lines, lines with the same slopes

• Infinitely Many Solutions: One line (Both of the lines are the same)

Multiple Choice

Solve:
y = 2x -11
-3y = -6x -15

• (-1.5,-4)
• (1.5,4)
• No solution (parallel lines)
• Infinitely many solutions

Multiple Choice

Solve:
y = 7x + 9
2y + 2x = -18

• (5, 0)
• (9, 18)
• (-2, -5)
• (1, 16)

Multiple Choice

Solve the following system by graphing.  What is the solution?

• Infinite number of solutions
• (3, 3)
• (3, -3)
• (-3, 3)

Multiple Choice

^{What is the solution?} 

• 1
• -2
• (1, 2)
• (1, -1)

Multiple Choice

When you graph the exact same equation twice,

• you will have no solution. 
• you will have one solution.
• you will have infinite solutions. 
• you will graph a giraffe. 

Multiple Choice

How many solutions will this system have? 

• No solution
• One Solution
• I Don't Know
• Infinitely Many Solutions

Elimination

• Best when the equation is in Standard Form

• Look for matching coefficients and variables that OPPOSITE signs

• If needed you can multiply by a constant to make variables match and be opposite

• 
  

Substitution

• Used when we we have a single variable isolated.

• The coefficient of the isolated variable should be 1.

• Replace the variable in 1 equation, with the isolated variable from the other equation.

• Find the other solution by replacing the unknown variable with the newly found variable.

Multiple Choice

What is the best method to use?

2x − 3y = −1

y =x − 1

• Graphing

• Substitution because a variable is defined

• Elimination because the equations are in standard form.

Multiple Choice

What is the best method to use?

−4x − 2y = −12

4x + 8y = −24

• Graphing

• Substitution because a variable is defined

• Elimination because the equations are in standard form.

Multiple Choice

What variable do you eliminate?

4x + 8y = 20

−4x + 2y = −30

• X because they have opposite signs

• Y because they have opposite signs

Multiple Choice

What is the proper way to set up the equation to begin solving for x?

y = 6x − 11

−2x − 3y = −7

• -2(-11) -3y= -7

• -2(6x-11) -3y = -7

• -2x -3(11) = -7

• -2x - 3(6x-11)= -7

Multiple Choice

What is the best method to use?

−4x − 2y = −12

4x + 8y = −24

• Graphing

• Substitution because a variable is defined

• Elimination because the equations are in standard form.

Multiple Choice

Which variable do you eliminate and why?

7x + 2y = 24

8x + 2y = 30

• You pick x because it's LCM is 56

• You pick y because it's LCM is 2.

Multiple Choice

What variable do you eliminate, and what do you multiply the equation(s) by?

5x + y = 9

10x − 7y = −18

• You eliminate x, and multiply the top equation by -2

• You eliminate y, and multiply the top equation by 7

Multiple Choice

What is the best method do you use?

y = 6x − 11

−2x − 3y = −7

• Graphing

• Substitution because a variable is defined

• Elimination because both equations are in standard form.

Writing Systems of Equations

• Identify the variables

• Look for slopes and starting (y-intercepts)

• Write the equation

• Solve with Desmos

Multiple Choice

Some students want to order shirts with their school logo. One company charges $9.65 per shirt plus a setup fee of $43. Another company charges $8.40 per shirt plus a $58 fee. Which equation represents the number of shirts when both companies charge the same amount? 

• y = 9.65 + x
  y = 8.40 + x
• y = 9.65x + 43
  y = 8.40x + 58
• y =9.65x
  y = 8.40x
• y = 9.65x - 43
  y = 8.40x - 58

Multiple Choice

A large pizza at Palanzio’s Pizzeria costs $6.80 plus $0.90 for each topping. The cost of a large cheese pizza at Guido’s Pizza is $7.30 plus $0.65 for each topping. Which system of equations could be used to find the number of toppings when both companies cost the same amount? 

• y = 6.80 + .65x
  y=7.30+.90x
• x + y = 6.80
  x + y = 7.30
• y = 6.80+.90x
  y = 7.30 + .65x
• y + .90x = 6.80
  y + .65x = 7.30

Multiple Choice

Two brothers went shopping at a back-to-school sale where all shirts and shorts were the same price. The younger brother spent $175 on 7 new shirts and 7 pairs of shorts. The older brother purchased 6 new shirts and 7 pairs of shorts and paid a total of $165. How much did one shirt cost?

• $5
• $10
• $15
• $20