Math Problem Solver

The system of linear equations has exactly one solution.

The substitutions are \( x = - \frac{56}{81} \), \( y = - \frac{88}{81} \).

\[y - x = - \frac{32}{81}\]

Solve \(4 x - 5 y = \frac{8}{3}\) for \(x\). \[x = \frac{5 y}{4} + \frac{2}{3}\]

Substitute \(\frac{5 y}{4} + \frac{2}{3}\) for \(x\) in \(66 x = 42 y\) and simplify. $$\begin{aligned}66 x &= 42 y \\ 66 \left(\frac{5 y}{4} + \frac{2}{3}\right) &= 42 y \\ y &= - \frac{88}{81} \end{aligned}$$

Use substitution of the numerical value of \(y\) to get the values of \(x\). $$\begin{aligned}x &= \frac{5 y}{4} + \frac{2}{3} \\ x &= 5 \left(-88\right) \frac{1}{81} \cdot \frac{1}{4} + \frac{2}{3} \\ x &= - \frac{56}{81}\end{aligned}$$

By substituting these values into the expression, we find that:$$\begin{aligned} y - x &= - \frac{88}{81} - - \frac{56}{81}\\ &=- \frac{32}{81}\end{aligned}$$