Question

determine the number of solutions to this system of linear equations y=x+5 y=−x+5

Answer 1

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

\[
y = x + 5
\]

\[
y = -x + 5
\]

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

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

Now, we can solve for \(x\). First, add \(x\) to both sides:

\[
x + x + 5 = 5
\]

This simplifies to:

\[
2x + 5 = 5
\]

Next, subtract 5 from both sides:

\[
2x = 0
\]

Now divide by 2:

\[
x = 0
\]

Now, we can find \(y\) by substituting \(x = 0\) into either of the original equations. Using the first equation:

\[
y = 0 + 5 = 5
\]

So, the solution to the system of equations is:

\[
(x, y) = (0, 5)
\]

Since the two lines intersect at exactly one point, the system has exactly **one solution**.