To find the point of intersection for the lines given by the equations [tex]\( y = -2x + 5 \)[/tex] and [tex]\( y = x - 4 \)[/tex], we need to solve for the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] where both equations are satisfied simultaneously.
### Step-by-Step Solution:
1. Set the equations equal to each other:
Since both equations equal [tex]\( y \)[/tex], we set the right-hand sides equal to each other:
[tex]\[
-2x + 5 = x - 4
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
Combine like terms to isolate [tex]\( x \)[/tex]:
[tex]\[
-2x + 5 = x - 4
\][/tex]
Add [tex]\( 2x \)[/tex] to both sides:
[tex]\[
5 = 3x - 4
\][/tex]
Add [tex]\( 4 \)[/tex] to both sides:
[tex]\[
9 = 3x
\][/tex]
Divide both sides by [tex]\( 3 \)[/tex]:
[tex]\[
x = 3
\][/tex]
3. Substitute [tex]\( x \)[/tex] back into one of the original equations to find [tex]\( y \)[/tex]:
Use the equation [tex]\( y = x - 4 \)[/tex]:
[tex]\[
y = 3 - 4 = -1
\][/tex]
So, the coordinates of the point where the two lines intersect are [tex]\( (3, -1) \)[/tex].
Hence, the correct answer is:
[tex]\[
\boxed{(3, -1)}
\][/tex]
