Question

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 to the given system of linear equations:

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

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

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

Now, solve for \( x \):

1. Add \( 2x \) to both sides:
\[
3x + 2x - 7 = 5
\]
\[
5x - 7 = 5
\]

2. Add \( 7 \) to both sides:
\[
5x = 12
\]

3. Divide by \( 5 \):
\[
x = \frac{12}{5}
\]

Next, substitute \( x = \frac{12}{5} \) back into either equation to find \( y \). Let's use the first equation:

\[
y = 3\left(\frac{12}{5}\right) - 7
\]
\[
y = \frac{36}{5} - 7
\]
\[
y = \frac{36}{5} - \frac{35}{5}
\]
\[
y = \frac{1}{5}
\]

Thus, the solution to the system is:

\[
\left( \frac{12}{5}, \frac{1}{5} \right)
\]

Since we found a unique solution, the number of solutions to this system of linear equations is **1**.