To find the solution to the system of equations \( y = \frac{2}{3}x - 5 \) and \( y = \frac{7}{3}x - 1 \), we can set the two equations equal to each other since they both equal \( y \).
\[ \frac{2}{3}x - 5 = \frac{7}{3}x - 1 \]
Now, let's solve for \( x \):
- Start by eliminating the fractions. We can do this by multiplying every term by 3:
\[ 3\left(\frac{2}{3}x\right) - 3(5) = 3\left(\frac{7}{3}x\right) - 3(1) \]
This simplifies to:
\[ 2x - 15 = 7x - 3 \]
- Next, isolate \( x \) by getting all the terms involving \( x \) on one side and the constant terms on the other side:
\[ 2x - 7x = -3 + 15 \]
This simplifies to:
\[ -5x = 12 \]
- Now, divide by -5:
\[ x = -\frac{12}{5} \]
- Substitute \( x \) back into one of the original equations to find \( y \). We'll use \( y = \frac{2}{3}x - 5 \):
\[ y = \frac{2}{3} \left(-\frac{12}{5}\right) - 5 \]
This is simplified as follows:
\[ y = -\frac{24}{15} - 5 \]
Convert -5 into a fraction with a denominator of 15:
\[ y = -\frac{24}{15} - \frac{75}{15} = -\frac{99}{15} \]
Thus, the solution to the system of equations is:
\[ \left(-\frac{12}{5}, -\frac{99}{15}\right) \]
To check if this point or any other point is the solution, substitute it back into the original equations. If the calculations for different points are requested, please provide those points and I can help determine if any of them are solutions.