Which of the following uses the substitution method to solve linear-quadratic system y=x2−4x+8 , y=3x−4 ?(1 point) Responses (−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) left parenthesis 4 comma 8 right parenthesis (4,8) and (3,5)

Question

GPT-4o mini · Answer

To solve the system of equations using the substitution method for the equations $y = x^{2} - 4 x + 8$ and $y = 3 x - 4$ , we can set the two equations equal to each other since they both equal $y$ .

Set the equations equal:

$x^{2} - 4 x + 8 = 3 x - 4$

Rearranging the equation gives:

$x^{2} - 4 x - 3 x + 8 + 4 = 0$ $x^{2} - 7 x + 12 = 0$

Next, we can factor the quadratic:

$(x - 3) (x - 4) = 0$

Setting each factor to zero gives us the values of $x$ :

$x - 3 = 0 \Rightarrow x = 3$ $x - 4 = 0 \Rightarrow x = 4$

Now we substitute back to find $y$ for both values of $x$ :

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, another solution is $(4, 8)$ .

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

The correct response is: $(4, 8)$ and $(3, 5)$ .