Question
Use the elimination method to solve the linear-quadratic system y=x2+10
, y=−7x−2
.(1 point)
Responses
(−3,19)
and (−4,26)
left parenthesis negative 3 comma 19 right parenthesis and left parenthesis negative 4 comma 26 right parenthesis
(−4,26)
left parenthesis negative 4 comma 26 right parenthesis
(−3,19)
left parenthesis negative 3 comma 19 right parenthesis
(3,−23)
and (4,−30)
, y=−7x−2
.(1 point)
Responses
(−3,19)
and (−4,26)
left parenthesis negative 3 comma 19 right parenthesis and left parenthesis negative 4 comma 26 right parenthesis
(−4,26)
left parenthesis negative 4 comma 26 right parenthesis
(−3,19)
left parenthesis negative 3 comma 19 right parenthesis
(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. \( y = -7x - 2 \)
Since both equations equal \( y \), we can set them equal to each other:
\[
x^2 + 10 = -7x - 2
\]
Now, rearranging the equation to set it to zero:
\[
x^2 + 7x + 12 = 0
\]
Next, we can factor the quadratic equation:
\[
(x + 3)(x + 4) = 0
\]
Setting each factor to zero gives us the solutions for \( x \):
1. \( x + 3 = 0 \) ⟹ \( x = -3 \)
2. \( x + 4 = 0 \) ⟹ \( x = -4 \)
Now, we will substitute these \( x \) values back into one of the original equations to find the corresponding \( y \) values. We can use \( y = -7x - 2 \).
**For \( x = -3 \):**
\[
y = -7(-3) - 2 = 21 - 2 = 19
\]
So one solution is \( (-3, 19) \).
**For \( x = -4 \):**
\[
y = -7(-4) - 2 = 28 - 2 = 26
\]
So the other solution is \( (-4, 26) \).
Thus, the solutions to the system are:
\[
(-3, 19) \text{ and } (-4, 26)
\]
The correct response is:
**(−3,19) and (−4,26)**.
1. \( y = x^2 + 10 \)
2. \( y = -7x - 2 \)
Since both equations equal \( y \), we can set them equal to each other:
\[
x^2 + 10 = -7x - 2
\]
Now, rearranging the equation to set it to zero:
\[
x^2 + 7x + 12 = 0
\]
Next, we can factor the quadratic equation:
\[
(x + 3)(x + 4) = 0
\]
Setting each factor to zero gives us the solutions for \( x \):
1. \( x + 3 = 0 \) ⟹ \( x = -3 \)
2. \( x + 4 = 0 \) ⟹ \( x = -4 \)
Now, we will substitute these \( x \) values back into one of the original equations to find the corresponding \( y \) values. We can use \( y = -7x - 2 \).
**For \( x = -3 \):**
\[
y = -7(-3) - 2 = 21 - 2 = 19
\]
So one solution is \( (-3, 19) \).
**For \( x = -4 \):**
\[
y = -7(-4) - 2 = 28 - 2 = 26
\]
So the other solution is \( (-4, 26) \).
Thus, the solutions to the system are:
\[
(-3, 19) \text{ and } (-4, 26)
\]
The correct response is:
**(−3,19) and (−4,26)**.