Solving Simultaneous Equations Using Substitution Quick Check

Question

4 of 54 of 5 Items
Question
Which of the following uses the substitution method to solve linear-quadratic system y=x2−4x+8, y=3x−4?(1 point)
Responses

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

(4,8)
left parenthesis 4 comma 8 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
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Answer 1

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

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

We can set the two equations equal to each other since both equal \( y \):

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

Now, let's rearrange it to form a standard quadratic equation:

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

Now, factor the quadratic:

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

Setting each factor to zero gives us the values for \( x \):

1. \( x - 3 = 0 \) → \( x = 3 \)
2. \( x - 4 = 0 \) → \( x = 4 \)

Next, we can find the corresponding \( y \) values by substituting \( x \) back into either of the original equations. We'll use the second equation for simplicity:

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

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

Thus, the solutions to the system are:

• \( (3, 5) \)
• \( (4, 8) \)

The correct response is:

(4,8) and (3,5)

or in the requested format:

left parenthesis 4 comma 8 right parenthesis and left parenthesis 3 comma 5 right parenthesis.