# How do you solve 6x+5y=10 and 7x-3y=24?

Aug 12, 2017

There are two common ways:

• Substitution
• Elimination

I can show both ways, but in this case I think elimination would be easier.

$6 x + 5 y = 10$
$7 x - 3 y = 24$

I got:

$\left(x , y\right) = \left(\frac{150}{53} , - \frac{74}{53}\right)$

SUBSTITUTION

Solve for one variable first.

$6 x = 10 - 5 y$

$\implies x = \frac{5}{3} - \frac{5}{6} y$

Plug it into the second equation to solve for the other variable.

$7 \left(\frac{5}{3} - \frac{5}{6} y\right) - 3 y = 24$

$\frac{35}{3} - \frac{35}{6} y - 3 y = 24$

Multiply through by $6$ to make this look nicer.

$70 - 35 y - 18 y = 144$

$- 53 y = 144 - 70$

$\textcolor{g r e e n}{y} = \frac{144 - 70}{- 53} = \textcolor{g r e e n}{- \frac{74}{53}}$

Therefore, for $x$ we get:

$\textcolor{g r e e n}{x} = \frac{5}{3} - \frac{5}{6} \left(- \frac{74}{53}\right)$

$= \frac{5}{3} + \frac{370}{318} = \frac{5}{3} + \frac{185}{159}$

$= \frac{795}{477} + \frac{555}{477} = \frac{1350}{477}$

$= \textcolor{g r e e n}{\frac{150}{53}}$

So, apparently, $\textcolor{b l u e}{\left(x , y\right) = \left(\frac{150}{53} , - \frac{74}{53}\right)}$... Let's check our answer.

6(150/53) + 5(-74/53) stackrel(?" ")(=) 10

900/53 - 370/53 stackrel(?" ")(=) 10 => 530/53 = 10 color(blue)(sqrt"")

7(150/53) - 3(-74/53) stackrel(?" ")(=) 24

1050/53 + 222/53 stackrel(?" ")(=) 24 => 1272/53 = (12 cdot 106)/53 = 24 color(blue)(sqrt"")

Yep, it's right! Wow, not nice-looking at all!

ELIMINATION

In this case we would be scaling one of the equations with fractions to eliminate a variable.

$\text{ } \frac{3}{5} \left(6 x + \cancel{5 y} = 10\right)$
$+ \text{ } 7 x - \cancel{3 y} = 24$
$\overline{\text{ "" "" "" "" "" "" "" }}$
$\text{ } \frac{18}{5} x + 7 x = 30$

$\left(\frac{18}{5} + \frac{35}{5}\right) x = 30$

$\textcolor{g r e e n}{x} = \frac{30}{\frac{18}{5} + \frac{35}{5}}$

$= \frac{30}{\frac{53}{5}} = \textcolor{g r e e n}{\frac{150}{53}}$

And now, plug it into the second equation.

$7 \left(\frac{150}{53}\right) - 3 y = 24$

$7 \left(\frac{150}{53}\right) - 24 = 3 y$

$\textcolor{g r e e n}{y} = \frac{7}{3} \left(\frac{150}{53}\right) - 8$

$= 7 \left(\frac{50}{53}\right) - \frac{424}{53}$

$= \frac{350}{53} - \frac{424}{53}$

$= \textcolor{g r e e n}{- \frac{74}{53}}$

And as before, we get:

$\textcolor{b l u e}{\left(x , y\right) = \left(\frac{150}{53} , - \frac{74}{53}\right)}$

So if you get something ugly like this, it's right!