Question

Determine the number of solutions to this system of linear equations (0, 1, ∞)
:
y=3x−7

y=−2x+5
(1 point)
This system of linear equations has
solution(s).

Answer 1

To determine the number of solutions for the system of equations given by:

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

we can set them equal to each other since both expressions equal \( y \):

\[
3x - 7 = -2x + 5
\]

Now, let's solve for \( x \):

1. First, add \( 2x \) to both sides:

\[
3x + 2x - 7 = 5
\]

Simplifying this gives:

\[
5x - 7 = 5
\]

2. Next, add \( 7 \) to both sides:

\[
5x = 12
\]

3. Now, divide by \( 5 \):

\[
x = \frac{12}{5}
\]

Now that we have \( x \), we can substitute it back into either equation to find \( y \). Using the first equation:

\[
y = 3\left(\frac{12}{5}\right) - 7
\]

Calculating this gives:

\[
y = \frac{36}{5} - 7 = \frac{36}{5} - \frac{35}{5} = \frac{1}{5}
\]

So the solution is \( \left(\frac{12}{5}, \frac{1}{5}\right) \).

Since we have found one unique solution, the system of linear equations has **1 solution**.

Thus, the system of linear equations has **1** solution.