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#?

Question

Sadie Alexander · Accepted Answer

#color(red)(x=5/19,y=46/19)#

One way is to use Cramer's Rule.

Step 1. Enter your equations.

#y=1/3x+7/3# #y=-5/4x+11/4#

Step 2. Write them in standard form.

#1/3x-y=-7/3# #5/4x+y=11/4#

Step 3. Multiply to get rid of fractions.

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

The left hand side gives us the coefficient matrix.

#((1,-3),(5,4))#

The right hand side gives us the answer matrix.

#((-7),(11))#

The determinant #D# of the coefficient matrix is

#D = |(1,-3),(5,4)| = 4+15=19#

Let #D_x# be the determinant formed by replacing the #x#-column values with the answer-column values:

#D_x=|(-7,-3),(11,4)| = -28+33 = 5#

Similarly,

#D_y=|(1,-7),(5,11)|=11+35=46#

Cramer's Rule says that

#x = D_x/D =5/19#,

#y = D_y/D=46/19#,

The solution is #x=5/19,y=46/19#

Check:

#y=1/3x+7/3#

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

#46/19=46/19#

#y=-5/4x+11/4#

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

#46/19=46/19#

It works!

Our solution is correct.

Cora Alexander · Answer

undefined To use matrices to find the solution of the system of equations $y = \frac{1}{3}x + \frac{7}{3}$ and $y = -\frac{5}{4}x + \frac{11}{4}$, you first need to set up a matrix equation. Represent the coefficients of $x$, $y$, and the constants on one side of the equation and the variables on the other side. Then, use matrix operations to solve for the variables.

The system of equations can be rewritten as:

$$
\begin{align*}
\frac{1}{3}x - y &= -\frac{7}{3}\
-\frac{5}{4}x - y &= -\frac{11}{4}
\end{align*}
$$

This can be represented in matrix form as:

$$
\begin{bmatrix}
\frac{1}{3} & -1\
-\frac{5}{4} & -1
\end{bmatrix}
\begin{bmatrix}
x\
y
\end{bmatrix}
=
\begin{bmatrix}
-\frac{7}{3}\
-\frac{11}{4}
\end{bmatrix}
$$

To solve for $x$ and $y$, you can use the inverse of the coefficient matrix. First, find the inverse of the coefficient matrix:

$$
\begin{bmatrix}
\frac{1}{3} & -1\
-\frac{5}{4} & -1
\end{bmatrix}^{-1}
$$

Then, multiply the inverse of the coefficient matrix by the constant matrix:

$$
\begin{bmatrix}
x\
y
\end{bmatrix}
=
\begin{bmatrix}
\frac{1}{3} & -1\
-\frac{5}{4} & -1
\end{bmatrix}^{-1}
\begin{bmatrix}
-\frac{7}{3}\
-\frac{11}{4}
\end{bmatrix}
$$

After finding the product, you will have the values of $x$ and $y$, which represent the solution to the system of equations.

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#?

[ \begin{bmatrix} \frac{1}{3} & -1\ -\frac{5}{4} & -1 \end{bmatrix} \begin{bmatrix} x\ y \end{bmatrix}

[ \begin{bmatrix} x\ y \end{bmatrix}