Question

Determine the number of solutions to this system of linear equations: y=x+5 y=−x+5 (1 point)

1
1
2
2
0
0
infinite

Answer 1

To determine the number of solutions to the system of linear equations given by:

1. \( y = x + 5 \)
2. \( y = -x + 5 \)

we can analyze the two equations graphically.

1. The first equation \( y = x + 5 \) is a straight line with a slope of 1 and a y-intercept at (0, 5). This means it rises to the right and crosses the y-axis at (0, 5).

2. The second equation \( y = -x + 5 \) is also a straight line, but with a slope of -1 and the same y-intercept at (0, 5). This line falls to the right and also crosses the y-axis at (0, 5).

To find the intersection, we can set the two equations equal to each other:

\[
x + 5 = -x + 5
\]

Solving for \( x \):

\[
x + x = 5 - 5
\]
\[
2x = 0
\]
\[
x = 0
\]

Now, substituting \( x = 0 \) back into either equation to find \( y \):

Using the first equation:
\[
y = 0 + 5 = 5
\]

So, the two lines intersect at the point (0, 5).

Since the two lines have different slopes (1 and -1), they are not parallel and will intersect at exactly one point.

Thus, the system of equations has exactly **one solution**:
\[
\boxed{1}
\]