Answer
$x = -1$
$y = -2$
There is only one solution to this system of equations.
Work Step by Step
To solve a system of equations using substitution, we plug in an expression in place of a variable.
We want to modify one of the equations so that one variable is expressed in terms of another. We can then use this equation to substitute for the variable in the other equation.
Let's modify the second equation:
$x - 3y = 5$
Add $3y$ to each side of the equation to isolate $x$:
$x = 3y + 5$
Substitute this expression for $x$ into the first equation:
$\frac{1}{3}y = \frac{7}{3}(3y + 5) + \frac{5}{3}$
Use the distributive property on the right side of the equation:
$\frac{1}{3}y = \frac{21y}{3} + \frac{35}{3} + \frac{5}{3}$
Combine like terms on the right side of the equation:
$\frac{1}{3}y = \frac{21y}{3} + \frac{40}{3}$
Subtract $\frac{21y}{3}$ from each side of the equation:
$-\frac{20}{3}y = \frac{40}{3}$
Divide each side of the equation by $-\frac{20}{3}$ to solve for $y$. Dividing by a fraction is the same as multiplying by its reciprocal:
$y = -\frac{3}{20}(\frac{40}{3})$
Multiply the two fractions:
$y = -\frac{120}{60}$
Simplify the fraction by dividing both the numerator and denominator by their greatest common factor, $60$:
$y = -2$
Now that we have the value for $y$, we can substitute it into the second equation to solve for $x$:
$x - 3(-2) = 5$
Multiply first:
$x + 6 = 5$
Subtract $6$ on both sides of the equation to solve for $x$:
$x = -1$
There is only one solution to this system of equations.
