Solving Systems of Equations

Solving Systems of Equations

Methods We Have Used to Solve Systems of Equations

• ​Substitution

• Elimination

• ​Graphing

Lets take a look!

One Solution looks like..

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• ​Two lines intersect at one point

• ​Solution is an ordered pair (x,y)

No Solution Looks Like..

• These lines will never intersect

• ​Parallel Lines

• ​Same slopes different y-intercepts

Infinite Amount of Solutions Looks Like..

Lines on top of each other

• ​They are the same line when simplified or written in slope-intercept form

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.

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

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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 Select

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

Solving by Substitution

Steps 1 & 2

Solving by Substitution

Steps 3 & 4

The solution is (42,-12)

Solving by Substitution

Example 2

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 solution to the system of equations?

y = -2x + 18

y = 8

• (5, 8)
• (-5, 8)
• (2, 8)
• (-2, 8)

Multiple Choice

Solve the system by substitution.  

5x + 4y= −14

y =  −7x  −  15 

• (-2, -1)
• (1, -2)
• (-2, 1)
• (-1, -2)

Solving by Elimination

Steps 1 & 2

Solving by Elimination

Steps 3/4 & 5/6

The solution is (2,4)

Solving by Elimination

Example 2

Multiple Choice

Solve using Elimination.   

4x + 8y =  20

-4x + 2y = -30

• (-7,1)
• (2,-5)
• (-2,5)
• (7,-1)

Multiple Choice

Solve by elimination   

2x + 9y = -7

6x - 3y = 9 

• (-1, -1) 
• (2,-1)
• (1,1)
• (1,-1)

Multiple Choice

What is the solution to this system of equations?

• (4, -1)
• (-1, 4)
• (-4, 1)
• (-4, -1)

Multiple Choice

What is the solution?
(Hint:  Both lines are graphed over each other)

• One Solution
• No solution
• Infinitely Many Solutions

Multiple Choice

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

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

Multiple Choice

Solve by graphing:

y=8x+1

y=6x+3

• (1, 9)
• (4, 14)
• (0, 1)
• (2, 4)