To determine the number of solutions for the system of equations given by:
- \( y = 3x - 7 \)
- \( y = -2x + 5 \)
we can set them equal to each other since both expressions equal \( y \):
\[ 3x - 7 = -2x + 5 \]
Now, let's solve for \( x \):
- First, add \( 2x \) to both sides:
\[ 3x + 2x - 7 = 5 \]
Simplifying this gives:
\[ 5x - 7 = 5 \]
- Next, add \( 7 \) to both sides:
\[ 5x = 12 \]
- 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.