To determine the [tex]\( y \)[/tex]-coordinate of the intersection of the two equations [tex]\( y = 5x - 9 \)[/tex] and [tex]\( y = x^2 - 3x + 7 \)[/tex], we need to find a common [tex]\( x \)[/tex] value that satisfies both equations. Here is the step-by-step solution:
1. Set the equations equal to each other: Since [tex]\( y \)[/tex] is the same in both equations at the point of intersection, we equate the two expressions for [tex]\( y \)[/tex]:
[tex]\[
5x - 9 = x^2 - 3x + 7
\][/tex]
2. Rearrange to form a quadratic equation: Move all terms to one side to set the equation to zero:
[tex]\[
x^2 - 3x + 7 - 5x + 9 = 0 \implies x^2 - 8x + 16 = 0
\][/tex]
3. Solve the quadratic equation: Factorize the equation [tex]\( x^2 - 8x + 16 = 0 \)[/tex]:
[tex]\[
(x - 4)(x - 4) = 0 \implies (x - 4)^2 = 0 \implies x = 4
\][/tex]
This quadratic equation has a double root [tex]\( x = 4 \)[/tex].
4. Find the corresponding [tex]\( y \)[/tex]-coordinate: Substitute [tex]\( x = 4 \)[/tex] back into either of the original equations to find [tex]\( y \)[/tex]. Using the first equation [tex]\( y = 5x - 9 \)[/tex]:
[tex]\[
y = 5(4) - 9 \implies y = 20 - 9 \implies y = 11
\][/tex]
Thus, the [tex]\( y \)[/tex]-coordinate of the solution is:
[tex]\[
y = 11.0
\][/tex]
Therefore, the [tex]\( y \)[/tex]-coordinate of the intersection of the two given equations is:
[tex]\[
y = 11
\][/tex]
