Question

Solve the following system of equations algebraically:

y = x2 - 7x - 5

y = 2x - 5

Answer 1

(0,-5) and (9,13)

Explanation:

Step 1: Set up

• You can solve system of equations algebraically either by elimination or substitution.
• You know there is going to be two solutions because of x²

Step 2: Solve

Substitution:

• 
  `y=x^(2) -7x-5`
• 
  `y=2x-5`
• Since both equations have been solved for y, you can just set them equal to each other.

1. 
   `2x-5=x^(2) -7x-5`
2. 
   `9x-x^(2) =0`
3. 
   `-x(x-9)=0`
4. 
   `x=0\\x=9`

• Then plug both x values into one of the equations to get their corresponding y values.

1. 
   `y=2(0)-5=-5`
2. 
   `y=2(9)-5=18-5=13`

Elimination:

• 
  `y=x^(2) -7x-5`
• 
  `y=2x-5`
• Since both equations have been solved for y, multiply one equation by -1 in order to cancel the y variable and solve for x.

1. 
   `y=x^(2) -7x-5`
2. 
   `-y=-2x+5`
3. 
   `0=x^(2) -9x+0`
4. 
   `0=x^(2) -9x`
5. 
   `0=x(x-9)`
6. 
   `x=0\\x=9`

• Then plug both x values into one of the equations to get their corresponding y values.

1. 
   `y=2(0)-5=-5`
2. 
   `y=2(9)-5=18-5=13`

Answer 2

x = 0; y = -5

x = 9; y = 13

Explanation:

y = x² - 7x - 5

y = 2x - 5

Use substitution to set the two right sides equal.

x² - 7x - 5 = 2x - 5

x² - 9x = 0

x(x - 9) = 0

x = 0 or x - 9 = 0

x = 0 or x = 9

Now for each x value, we find a corresponding y vlaue.

x = 0

y = 2x - 5 = 2(0) - 5 = -5

x = 9

y = 2x - 5 = 2(9) - 5 = 13

Answer:

x = 0; y = -5

x = 9; y = 13