Question

1. for the following questions, use the system of equations (1 point each):

3x + 2y = 14
2x - 4y = 4
a. solve the system of equations using either the substitution method or the multiplication/addition method.
b. check your solution by writing the system as a matrix equation and using the inverse matrix.
c. verify your solution by graphing the system. mark the point on your graph that represents the solution.

Explanation:

Step1: Solve using multiplication/addition method

Multiply the first equation $3x + 2y=14$ by 2 to get $6x + 4y = 28$. Add this to the second - equation $2x-4y = 4$.
$$(6x + 4y)+(2x-4y)=28 + 4$$
$$8x=32$$
$$x = 4$$

Step2: Find the value of y

Substitute $x = 4$ into the first equation $3x+2y = 14$.
$$3\times4+2y=14$$
$$12+2y=14$$
$$2y=14 - 12$$
$$2y=2$$
$$y = 1$$

Step3: Write the system as a matrix equation

The system

$$\begin{cases}3x + 2y=14\\2x-4y = 4\end{cases}$$

can be written as

$$\begin{bmatrix}3&2\\2&-4\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}14\\4\end{bmatrix}$$

. Let $A=

$$\begin{bmatrix}3&2\\2&-4\end{bmatrix}$$

$, $X=

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

$ and $B=

$$\begin{bmatrix}14\\4\end{bmatrix}$$

$. The inverse of $A$, $A^{-1}=\frac{1}{3\times(-4)-2\times2}

$$\begin{bmatrix}-4&-2\\-2&3\end{bmatrix}$$

=\frac{1}{-12 - 4}

$$\begin{bmatrix}-4&-2\\-2&3\end{bmatrix}$$

=-\frac{1}{16}

$$\begin{bmatrix}-4&-2\\-2&3\end{bmatrix}$$

=

$$\begin{bmatrix}\frac{1}{4}&\frac{1}{8}\\\frac{1}{8}&-\frac{3}{16}\end{bmatrix}$$

$. Then $X = A^{-1}B$.

$$\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}\frac{1}{4}&\frac{1}{8}\\\frac{1}{8}&-\frac{3}{16}\end{bmatrix}\begin{bmatrix}14\\4\end{bmatrix}=\begin{bmatrix}\frac{1}{4}\times14+\frac{1}{8}\times4\\\frac{1}{8}\times14-\frac{3}{16}\times4\end{bmatrix}=\begin{bmatrix}\frac{14}{4}+\frac{4}{8}\\\frac{14}{8}-\frac{12}{16}\end{bmatrix}=\begin{bmatrix}\frac{14}{4}+\frac{2}{4}\\\frac{28 - 12}{16}\end{bmatrix}=\begin{bmatrix}4\\1\end{bmatrix}$$

Step4: Graph the equations

Rewrite the first equation $3x + 2y=14$ as $y=-\frac{3}{2}x + 7$. The y - intercept is 7 and the slope is $-\frac{3}{2}$. Rewrite the second equation $2x-4y = 4$ as $y=\frac{1}{2}x - 1$. The y - intercept is - 1 and the slope is $\frac{1}{2}$. The point of intersection of the two lines is $(4,1)$.

Answer:

a. $x = 4,y = 1$
b. The solution is verified using the inverse - matrix method as $x = 4,y = 1$.
c. The point of intersection of the two lines $y=-\frac{3}{2}x + 7$ and $y=\frac{1}{2}x - 1$ on the graph is $(4,1)$, which verifies the solution.