Question

Given the system of equations: 2x – y = –2 x = 14 + 2y What is the value of the system determinant? What is the value of the y−determinant? What is the value of the x−determinant? What is the solution to the system of equations?

Answer 1

ANSWER 1

The given system  of equations is:

$2x-y=-2$

$x=14+2y$

Let us rewrite the second system to obtain,

$2x-y=-2$

$x-2y=14$

We need to apply the Cramer's rule. So we write out the coefficient matrices to obtain:

$\left[\begin{array}{cc}2&-1\\1&-2\end{array}\right]$

The answer column is;

$\left[\begin{array}{c}-2&14\end{array}\right]$

The value of the y-determinant is denoted by $D_y$. To find this we replace the coefficient determinant with answer-column values in y-column to get,

$D_y=\left|\begin{array}{cc}2&-2\\1&14\end{array}\right|$

Recall that,

If $A=\left[\begin{array}{cc}a&b\\c&d\end{array}\right]$

then, $det_{A}=\left|\begin{array}{cc}a&b\\c&d\end{array}\right|=ad-bc$

This implies that,

$D_y=2\times14-1\times(-2)$

$D_y=28+2$

$D_y=30$

ANSWER 2

The value of the x-determinant is denoted by $D_x$. To find this we replace the coefficient determinant with answer-column values in x-column to get,

$D_x=\left|\begin{array}{cc}-2&-1\\14&-2\end{array}\right|$

This implies that,

$D_x=-2\times-2-14\times-1$

$D_x=4+14$

$D_x=18$

ANSWER 3

To find the solution to the system of equations.

We need to find the determinant of the coefficient matrix.

$D=\left|\begin{array}{cc}2&-1\\1&-2\end{array}\right|$

This implies that,

$D=2\times(-2)-1\times-1$

$D=-4+1$

$D=-3$

Cramer's rule says that,

$x=\frac{D_x}{D}$

$\Rightarrow x=\frac{18}{-3}$

$\Rightarrow x=-6$

and

$y=\frac{D_y}{D}$

$\Rightarrow y=\frac{30}{-3}$

$\Rightarrow y=-10$

Therefore the solution is $x=-6$ and  $y=-10$.

Answer 2

2x - y = -2

x = 14 + 2y

Plug in the x equation

2(14+2y) - y = -2

Distribute

28 + 4y - y = -2

Combine variables

28 + 3y = -2

Subtract 28 on both sides

3y = -30

Divide the y

y = -10