Question

y=5x+7
3y-15x=18

One solution, no solution, or infinite solutions

Answer 1

To determine whether the system of equations has one solution, no solutions, or infinite solutions, we can analyze the two given equations:

1. \( y = 5x + 7 \)
2. \( 3y - 15x = 18 \)

First, we can rearrange the second equation into slope-intercept form (i.e., \( y = mx + b \)).

Starting with the second equation:
\[
3y - 15x = 18
\]

Add \( 15x \) to both sides:
\[
3y = 15x + 18
\]

Now, divide every term by 3:
\[
y = 5x + 6
\]

Now we have two linear equations:
1. \( y = 5x + 7 \)
2. \( y = 5x + 6 \)

Next, we can analyze these two equations:
- The first equation has a slope of \( 5 \) and a y-intercept of \( 7 \).
- The second equation also has a slope of \( 5 \) but a y-intercept of \( 6 \).

Since both lines have the same slope but different y-intercepts, they are parallel and will never intersect. Therefore, the system of equations has **no solutions**.