To determine the number of solutions to the system of linear equations:
- \( y = 3x - 7 \)
- \( y = -2x + 5 \)
we can set the two equations equal to each other and solve for \( x \):
\[ 3x - 7 = -2x + 5 \]
Now, add \( 2x \) to both sides:
\[ 3x + 2x - 7 = 5 \]
This simplifies to:
\[ 5x - 7 = 5 \]
Next, add 7 to both sides:
\[ 5x = 12 \]
Now, divide both sides by 5:
\[ x = \frac{12}{5} \]
Now we can substitute \( x = \frac{12}{5} \) back into either of the original equations to find the corresponding \( y \)-value. Let’s use the first equation:
\[ y = 3\left(\frac{12}{5}\right) - 7 \]
Calculating this:
\[ y = \frac{36}{5} - 7 = \frac{36}{5} - \frac{35}{5} = \frac{1}{5} \]
Thus, the solution to the system is:
\[ \left(\frac{12}{5}, \frac{1}{5}\right) \]
Since we found exactly one solution, the system has one solution.
Therefore, this system of linear equations has 1 solution.