Systems of Equations Review

Systems of Equations REVIEW

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Solutions to Systems of Equations

• You will have ONE SOLUTION when the two graphs intersect.

• You will have NO SOLUTIONS when the graphs are PARALLEL LINES. Or, when using the elimination method you get a FALSE statement like 0 = 3.

• You will have INFINITE SOLUTIONS when the graphs are THE SAME (one on top of the other one). Or, when using the elimination method you get a TRUE statement like 0 = 0.

Multiple Choice

If a system of linear equations has ONE solution, what does this mean about the two lines?

• Parallel lines

• the same line

• Intersecting lines

Multiple Choice

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

• One Solution
• No solution
• Infinitely Many Solutions

Multiple Choice

This system has _____ solutions

• no
• 1
• 2
• Infinitely many

Graphing Method

• To solve a system of equations by using the GRAPHING METHOD:

• If the equations are in Slope-Intercept Form, graph them on a coordinate plane and see where the lines intersect.

• If they are not in Slope-Intercept Form, use DESMOS to graph it and see where the lines intersect.

Graphing Example 1

What is the solution to the system of equations:

y = 2x - 1

y = -1/3 + 6

Graphing Example 2

What is the solution to the system of equations:

2x+4y=-8

3x-2y=-4

Go to desmos.com and click on "Graphing Calculator." Then type the first equation in line 1 and the second equation in line 2. Your answer is where the graphs intersect. You should get (-2, -1)!

Multiple Choice

Use the GRAPHING METHOD to solve the system of equations?

2x - 8y = -2

5x + 3y = 18

• (-3, 4)

• (3. 1)

• (1, 3)

• (4, -3)

Multiple Choice

Use the GRAPHING METHOD to solve the system of equations?

-6x - 2y = 14

5x - y = -25

• (2, -3)

• (-3. 2)

• (-4, 5)

• (5, -4)

Substitution Method

• To solve a system of equations by using the SUBSTITUTION METHOD:

• Get one of the variables by itself in one of the equations (if there is not already a variable by itself).

• Substitute for that variable in the other equation and solve.

• Fill that answer back into the original equation and solve for the second variable.

Substitution

Example

What is the solution to the system of equations:

y = x + 9

3x + 8y = -5

Multiple Choice

What is the first step to solve the system of equations by using the SUBSTITUTION METHOD?

y = 3x

4x - 2y = -6

• fill in 3x in the other equation like this:

  4(3x) - 2y = -6

• fill in 3x in the other equation like this:

  4x - 2(3x) = -6

Multiple Choice

Solve the system of equations by using the SUBSTITUTION METHOD?

y = 3x

4x - 2y = -6

• (-3, 9)

• (9, -3)

• (3, 9)

• (9, 3)

Multiple Choice

What is the first step to solve the system of equations by using the SUBSTITUTION METHOD?

x = 2y - 1

2x - 3y = 4

• fill in 2y - 1 in the other equation like this:

  2(2y - 1) - 3y = 4

• fill in 2y - 1 in the other equation like this:

  2x - 3(2y - 1) = 4

Multiple Choice

Solve the system of equations by using the SUBSTITUTION METHOD?

x = 2y + 1

2x - 3y = 4

• (5, 2)

• (2, 5)

• (-2, -5)

• (-5, -2)

Elimination Method

• To solve a system of equations by using the ELIMINATION METHOD:

• Make sure the equations are lined up (x's first, then y's, then = , then constant).

• Get either the x's or y's to cancel each other (same coefficient but OPPOSITE signs; for example -2x and +2x).

• Note: If you have the same coefficients and the SAME signs (like -5y and -5y) simply multiply ONE of the equations by -1. That will basically change all of the signs in the entire equation.

Elimination Method ... continued

• Combine the equations from top to bottom which will eliminate one variable.

• Solve for the variable that was not eliminated.

• Now that you know what one variable is, fill it back into either of the original equations and solve for the second variable.

Elimination Example 1

What is the solution to the system of equations:

3x - 2y = 8

2x + 2y = 12

Because this system is "ready to go" the -2y and +2y will cancel right now! After combining up and down we get 5x = 20. Divide both sides by 5 and x = 4. Fill in 4 for x in the second equation and you get:

2(4) + 2y = 12

8 + 2y = 12 (subtract 8 from both sides)

2y = 4 (divide by 2 on both sides)

y = 2

So your answer is (4, 2)!

Elimination Example 2

Elimination Example 2 ... continued

-x - 3y = -7

6x + 3y = -3

After combining up and down we get 5x = -10. Divide both sides by 5 and x = -2. Fill in -2 for x in the second equation and you get:

6(-2) + 3y = -3

-12 + 3y = -3 (add 12 to both sides)

3y = 9 (divide by 2 on both sides)

y = 3

So your answer is (-2, 3)!

Multiple Choice

-2x + 3y = -7

2x + 5y = -1

Is this system of equations "ready to go" where we can eliminate a variable right now?

• Yes, it is ready

• No, we have to multiply one

  of the equations by -1

Multiple Choice

-2x + 3y = -7

2x + 5y = -1

What is the result after combining the equations up and down and eliminating the x?

• 2x - 8y = 6

• 8y = -8

• 4x = 8y

• 12y = -8

Multiple Choice

-2x + 3y = -7

2x + 5y = -1

What is the solution to this system of equations?

• (4, -4)

• (-2, 3)

• (2, -1)

• (1, 4)

Multiple Choice

5x - 6y = -8

5x + 4y = 22

Is this system of equations "ready to go" where we can eliminate a variable right now?

• Yes, it is ready

• No, we have to multiply one

  of the equations by -1

Multiple Choice

5x - 6y = -8

5x + 4y = 22

If we multiply the bottom equation by -1, what would it change to?

• 5x - 4y = 22

• -5x - 4y = -22

• -5x +4y = 22

• 5x + 4y = -22

Multiple Choice

After multiplying the bottom equation by -1, we now have:

5x - 6y = -8

-5x - 4y = -22

What will be the result after combining the equations up and down and eliminating the x?

• -10x = 2y

• -2y = -22

• -10y = -30

• -x + 2y = 30

Multiple Choice

5x - 6y = -8

5x + 4y = 22

What is the solution of this system of equations?

• (-3, -2)

• (3, 2)

• -2, -3)

• (2, 3)