To determine the number of solutions to the given system of linear equations:
- \( y = 3x - 7 \)
- \( y = -2x + 5 \)
we can set the two equations equal to each other since both equal \( y \):
\[ 3x - 7 = -2x + 5 \]
Now, solve for \( x \):
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Add \( 2x \) to both sides: \[ 3x + 2x - 7 = 5 \] \[ 5x - 7 = 5 \]
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Add \( 7 \) to both sides: \[ 5x = 12 \]
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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.
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