Question

Determine the number of solutions in this system. Explain your answer.
y = 5x + 7
y = 2x + 7

Answer 1

To determine the number of solutions in the given system of equations

1. \( y = 5x + 7 \)
2. \( y = 2x + 7 \)

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

\[
5x + 7 = 2x + 7.
\]

Next, we can rearrange this equation to isolate \( x \). First, subtract \( 2x \) from both sides:

\[
5x - 2x + 7 = 7
\]

which simplifies to

\[
3x + 7 = 7.
\]

Now, subtract 7 from both sides:

\[
3x = 0.
\]

Dividing both sides by 3 gives:

\[
x = 0.
\]

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

\[
y = 5(0) + 7 = 7.
\]

The solution to the system is \( (0, 7) \).

To determine the number of solutions, we can analyze the lines represented by the equations. The first equation has a slope of 5, and the second equation has a slope of 2. Since the slopes are different, the lines will intersect at exactly one point.

Therefore, the system of equations has **exactly one solution** at the point \( (0, 7) \).