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)
left parenthesis 3 comma negative 23 right parenthesis and left parenthesis 4 comma negative 30 right parenthesis
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)
left parenthesis 3 comma negative 23 right parenthesis and left parenthesis 4 comma negative 30 right parenthesis
Answers
GPT-4o mini
To solve the system of equations \( y = x^2 + 10 \) and \( y = -7x - 2 \) using the elimination method, we can set the two equations equal to each other, since they both represent \( y \).
1. Start with the two equations:
\( y = x^2 + 10 \)
\( y = -7x - 2 \)
2. Set them equal to each other:
\( x^2 + 10 = -7x - 2 \)
3. Rearrange the equation to bring all terms to one side:
\( x^2 + 7x + 12 = 0 \)
4. Factor the quadratic:
\( (x + 3)(x + 4) = 0 \)
5. Set each factor to zero:
\( x + 3 = 0 \) or \( x + 4 = 0 \)
This gives us the solutions:
\( x = -3 \) or \( x = -4 \)
6. Now substitute these \( x \) values back into one of the original equations to find the corresponding \( y \) values. We can use \( y = -7x - 2 \) for this purpose.
- For \( x = -3 \):
\[
y = -7(-3) - 2 = 21 - 2 = 19
\]
- For \( x = -4 \):
\[
y = -7(-4) - 2 = 28 - 2 = 26
\]
7. So the solutions to the system are:
\( (-3, 19) \) and \( (-4, 26) \).
Hence, the final answer is:
\((−3,19) \text{ and } (−4,26)\)
1. Start with the two equations:
\( y = x^2 + 10 \)
\( y = -7x - 2 \)
2. Set them equal to each other:
\( x^2 + 10 = -7x - 2 \)
3. Rearrange the equation to bring all terms to one side:
\( x^2 + 7x + 12 = 0 \)
4. Factor the quadratic:
\( (x + 3)(x + 4) = 0 \)
5. Set each factor to zero:
\( x + 3 = 0 \) or \( x + 4 = 0 \)
This gives us the solutions:
\( x = -3 \) or \( x = -4 \)
6. Now substitute these \( x \) values back into one of the original equations to find the corresponding \( y \) values. We can use \( y = -7x - 2 \) for this purpose.
- For \( x = -3 \):
\[
y = -7(-3) - 2 = 21 - 2 = 19
\]
- For \( x = -4 \):
\[
y = -7(-4) - 2 = 28 - 2 = 26
\]
7. So the solutions to the system are:
\( (-3, 19) \) and \( (-4, 26) \).
Hence, the final answer is:
\((−3,19) \text{ and } (−4,26)\)