Which of the following uses the substitution method to solve linear-quadratic system y=x^2−4x+8, y=3x−4?(1 point)

Question

Responses

(4,8) and (3,5)
left parenthesis 4 comma 8 right parenthesis and left parenthesis 3 comma 5 right parenthesis

(−4,−16) and (−3,−13)
left parenthesis negative 4 comma negative 16 right parenthesis and left parenthesis negative 3 comma negative 13 right parenthesis

(3,5)
left parenthesis 3 comma 5 right parenthesis

(4,8)

Answer 1

To solve the system of equations using the substitution method, we have:

1. \( y = x^2 - 4x + 8 \)
2. \( y = 3x - 4 \)

Since both expressions equal \( y \), we can set them equal to each other:

\[ x^2 - 4x + 8 = 3x - 4 \]

Now, let's rearrange this equation to bring everything to one side:

\[ x^2 - 4x - 3x + 8 + 4 = 0 \] \[ x^2 - 7x + 12 = 0 \]

Next, we can factor this quadratic equation:

\[ (x - 3)(x - 4) = 0 \]

Setting each factor to zero gives us the solutions:

\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \] \[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]

Now we will find the corresponding \( y \) values for each \( x \) using the second equation \( y = 3x - 4 \):

1. For \( x = 3 \): \[ y = 3(3) - 4 = 9 - 4 = 5 \] So one solution is \( (3, 5) \).

2. For \( x = 4 \): \[ y = 3(4) - 4 = 12 - 4 = 8 \] So the other solution is \( (4, 8) \).

Thus, the solutions to the system of equations are \( (4, 8) \) and \( (3, 5) \).

Based on the options you provided, the correct response is:

(4, 8) and (3, 5).