To solve the system of equations using the elimination method, we first rewrite both equations:
- \( y - x^2 = x - 42 \) becomes \( y = x^2 + x - 42 \)
- \( x + y = 6 \) can be rewritten as \( y = 6 - x \)
Now we can set the two expressions for \( y \) equal to each other:
\[ x^2 + x - 42 = 6 - x \]
Next, we simplify this equation:
\[ x^2 + x + x - 42 - 6 = 0 \]
\[ x^2 + 2x - 48 = 0 \]
Now, we can factor the quadratic:
\[ (x + 8)(x - 6) = 0 \]
This gives us:
\[ x + 8 = 0 \quad \text{or} \quad x - 6 = 0 \]
Which leads to:
\[ x = -8 \quad \text{or} \quad x = 6 \]
Now we use these values of \( x \) to find the corresponding values of \( y \):
-
For \( x = -8 \): \[ y = 6 - (-8) = 6 + 8 = 14 \]
-
For \( x = 6 \): \[ y = 6 - 6 = 0 \]
Thus, the solutions to the system are:
- \( (-8, 14) \)
- \( (6, 0) \)
Looking at the provided answer choices, the correct answer is:
1. (-8, 14) and (6, 0)
1 answer