# How do I use matrices to find the solution of the system of equations y=1/3x+7/3 and y=−5/4x+11/4?

Aug 16, 2015

$\textcolor{red}{x = \frac{5}{19} , y = \frac{46}{19}}$

#### Explanation:

One way is to use Cramer's Rule.

$y = \frac{1}{3} x + \frac{7}{3}$
$y = - \frac{5}{4} x + \frac{11}{4}$

Step 2. Write them in standard form.

$\frac{1}{3} x - y = - \frac{7}{3}$
$\frac{5}{4} x + y = \frac{11}{4}$

Step 3. Multiply to get rid of fractions.

$x - 3 y = - 7$
$5 x + 4 y = 11$

The left hand side gives us the coefficient matrix.

$\left(\begin{matrix}1 & - 3 \\ 5 & 4\end{matrix}\right)$

The right hand side gives us the answer matrix.

$\left(\begin{matrix}- 7 \\ 11\end{matrix}\right)$

The determinant $D$ of the coefficient matrix is

$D = | \left(1 , - 3\right) , \left(5 , 4\right) | = 4 + 15 = 19$

Let ${D}_{x}$ be the determinant formed by replacing the $x$-column values with the answer-column values:

${D}_{x} = | \left(- 7 , - 3\right) , \left(11 , 4\right) | = - 28 + 33 = 5$

Similarly,

${D}_{y} = | \left(1 , - 7\right) , \left(5 , 11\right) | = 11 + 35 = 46$

Cramer's Rule says that

$x = {D}_{x} / D = \frac{5}{19}$,

$y = {D}_{y} / D = \frac{46}{19}$,

The solution is $x = \frac{5}{19} , y = \frac{46}{19}$

Check:

$y = \frac{1}{3} x + \frac{7}{3}$

46/19=1/3(5/19)+7/3 =5/57 +7/3 =(5/57+7×19/57) = (5+133)/57=138/57

$\frac{46}{19} = \frac{46}{19}$

$y = - \frac{5}{4} x + \frac{11}{4}$

46/19=-5/4×5/19+11/4=-25/78+(11×19)/78=(-25+209)/78=184/78

$\frac{46}{19} = \frac{46}{19}$

It works!

Our solution is correct.