Solve the following system using elimination or substitution. show your work.
y = x^2 - 2x - 6
y = 4x + 10
Please explain how to do this and the answer! THANKS!
y = x^2 - 2x - 6
y = 4x + 10 subtract the second equation from the first:
0=x^2-6x-16 then factor
(x-8)*(x+2)=0
x=8, or x=-2
To solve this system of equations using elimination or substitution, we need to find the values of x and y that satisfy both equations simultaneously.
Let's start by using substitution.
Step 1: Solve one equation for one variable in terms of the other variable.
From the second equation, we can solve for y in terms of x:
y = 4x + 10
Step 2: Substitute the expression found in step 1 into the other equation.
Now we substitute the expression for y in the first equation:
x^2 - 2x - 6 = 4x + 10
Step 3: Simplify the resulting equation.
Rearrange the equation to group like terms:
x^2 - 2x - 4x - 16 = 0
x^2 - 6x - 16 = 0
Step 4: Solve the quadratic equation.
We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's use factoring:
(x - 8)(x + 2) = 0
Setting each factor equal to zero and solving for x:
x - 8 = 0 --> x = 8
x + 2 = 0 --> x = -2
Step 5: Substitute the x-values back into either equation to find the corresponding y-values.
For x = 8, we substitute it back into the second equation:
y = 4(8) + 10 = 42
So one solution is (x, y) = (8, 42).
For x = -2, we substitute it back into the second equation:
y = 4(-2) + 10 = 2
So another solution is (x, y) = (-2, 2).
Therefore, the solution to the system of equations is (x, y) = (8, 42) and (-2, 2).