Question

Use the elimination method to solve the following linear-quadratic system: y−x2=x−42 x+y=6 (1 point) Responses (8,−2) and (−6,0) left parenthesis 8 comma negative 2 right parenthesis and left parenthesis negative 6 comma 0 right parenthesis (6,0) left parenthesis 6 comma 0 right parenthesis (−8,14) left parenthesis negative 8 comma 14 right parenthesis (−8,14) and (6,0)

Answer 1

To solve the system of equations using the elimination method, let’s first write down the equations in a manageable form.

The system is:

1. \( y - x^2 = x - 42 \)
2. \( x + y = 6 \)

### Step 1: Rearranging the first equation

We can rearrange the first equation to express it in standard form:

\[
y = x - 42 + x^2
\]

### Step 2: Substitute from the second equation

From the second equation \( x + y = 6 \), we can express \( y \) in terms of \( x \):

\[
y = 6 - x
\]

### Step 3: Substitute \( y \) in the first equation

Now substitute \( y = 6 - x \) into the rearranged first equation:

\[
6 - x = x - 42 + x^2
\]

### Step 4: Rearranging the equation

Now, simplify and rearrange:

\[
6 - x = x - 42 + x^2
\]
\[
6 = 2x - 42 + x^2
\]
\[
x^2 - 2x - 36 = 0
\]

### Step 5: Factoring the quadratic equation

Now factor the quadratic:

\[
(x - 6)(x + 6) = 0
\]

Setting each factor to zero gives:

\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\]
\[
x + 6 = 0 \quad \Rightarrow \quad x = -6
\]

### Step 6: Finding corresponding \( y \) values

Now plug these \( x \) values back into the equation \( y = 6 - x \) to find the corresponding \( y \):

1. For \( x = 6 \):

\[
y = 6 - 6 = 0 \quad \Rightarrow \quad (6, 0)
\]

2. For \( x = -6 \):

\[
y = 6 - (-6) = 6 + 6 = 12 \quad \Rightarrow \quad (-6, 12)
\]

### Step 7: Final solutions

Thus, the solutions to the system of equations are:

1. \( (6, 0) \)
2. \( (-6, 12) \)

### Answer Selection

The answer choices do not have \( (-6, 12) \), but they include \( (6, 0) \).

So, the final solutions to this system are:

\[
(6, 0)
\]

And the closest option from your responses would be \( (6, 0) \).