How do you solve x / 2 - y / 3 = 5/6 and x / 5 - y / 4 = 29 / 10 using substitution?

Jul 16, 2016

$x = - 13 , y = - 22$

Explanation:

Before we can even think of solving, let's change the equations into a more user-friendly form!.

Multiply each by the LCM of the denominators to get rid of the fractions.

$\frac{\textcolor{m a \ge n t a}{6} \times x}{2} - \frac{\textcolor{m a \ge n t a}{6} \times y}{3} = \frac{\textcolor{m a \ge n t a}{6} \times 5}{6} \text{ and " } \frac{x}{5} - \frac{y}{4} = \frac{29}{10}$

$3 x - 2 y = 5 \text{ and } \frac{\textcolor{b l u e}{20} \times x}{5} - \frac{\textcolor{b l u e}{20} \times y}{4} = \frac{\textcolor{b l u e}{20} \times 29}{10}$

$3 x - 2 y = 5 \text{ and } 4 x - 5 y = 58$

Make one of the variables in one equation the subject.

$\textcolor{m a \ge n t a}{x = \frac{5 + 2 y}{3}} \text{ or } y = \frac{3 x - 5}{2}$

(one is not really easier than the other to substitute.)

$4 \left(\textcolor{m a \ge n t a}{\frac{5 + 2 y}{3}}\right) - 5 y = 58$

$\frac{\left(20 + 8 y\right)}{3} - 5 y = 58 \text{ } \times 3$

$20 + 8 y - 15 y = 174$

$20 - 174 = 7 y$

$- 154 = 7 y$

$y = - 22 \text{ now use this value to find x}$

$x = \frac{5 + 2 y}{3} \Rightarrow \text{ } x = \frac{5 + 2 \left(- 22\right)}{3}$

$x = - 13$

Check these values in the second equation to verify.