Question
Use the elimination method to solve the linear-quadratic system y=x2+10, y=−7x−2.(1 point)%0D%0AResponses%0D%0A%0D%0A(−3,19)%0D%0Aleft parenthesis negative 3 comma 19 right parenthesis%0D%0A%0D%0A(−3,19) and (−4,26)%0D%0Aleft parenthesis negative 3 comma 19 right parenthesis and left parenthesis negative 4 comma 26 right parenthesis%0D%0A%0D%0A(−4,26)%0D%0Aleft parenthesis negative 4 comma 26 right parenthesis%0D%0A%0D%0A(3,−23) and (4,−30)
Answers
GPT-4o mini
To solve the system of equations using the elimination method, we have:
1. \( y = x^2 + 10 \)
2. \( y = -7x - 2 \)
First, we can set the two equations for \( y \) equal to each other:
\[
x^2 + 10 = -7x - 2
\]
Next, we rearrange the equation to bring all terms to one side:
\[
x^2 + 7x + 10 + 2 = 0
\]
This simplifies to:
\[
x^2 + 7x + 12 = 0
\]
Now, we will factor the quadratic:
\[
(x + 3)(x + 4) = 0
\]
Setting each factor to zero gives us the values of \( x \):
1. \( x + 3 = 0 \) → \( x = -3 \)
2. \( x + 4 = 0 \) → \( x = -4 \)
Now that we have the \( x \)-values, we can substitute them back into one of the original equations to find the corresponding \( y \)-values. We'll use \( y = -7x - 2 \) for this purpose.
For \( x = -3 \):
\[
y = -7(-3) - 2 = 21 - 2 = 19
\]
Thus, one solution is \( (-3, 19) \).
For \( x = -4 \):
\[
y = -7(-4) - 2 = 28 - 2 = 26
\]
Thus, the other solution is \( (-4, 26) \).
So the solutions to the system of equations are:
1. \( (-3, 19) \)
2. \( (-4, 26) \)
The correct response is:
**(−3,19) and (−4,26)**
1. \( y = x^2 + 10 \)
2. \( y = -7x - 2 \)
First, we can set the two equations for \( y \) equal to each other:
\[
x^2 + 10 = -7x - 2
\]
Next, we rearrange the equation to bring all terms to one side:
\[
x^2 + 7x + 10 + 2 = 0
\]
This simplifies to:
\[
x^2 + 7x + 12 = 0
\]
Now, we will factor the quadratic:
\[
(x + 3)(x + 4) = 0
\]
Setting each factor to zero gives us the values of \( x \):
1. \( x + 3 = 0 \) → \( x = -3 \)
2. \( x + 4 = 0 \) → \( x = -4 \)
Now that we have the \( x \)-values, we can substitute them back into one of the original equations to find the corresponding \( y \)-values. We'll use \( y = -7x - 2 \) for this purpose.
For \( x = -3 \):
\[
y = -7(-3) - 2 = 21 - 2 = 19
\]
Thus, one solution is \( (-3, 19) \).
For \( x = -4 \):
\[
y = -7(-4) - 2 = 28 - 2 = 26
\]
Thus, the other solution is \( (-4, 26) \).
So the solutions to the system of equations are:
1. \( (-3, 19) \)
2. \( (-4, 26) \)
The correct response is:
**(−3,19) and (−4,26)**