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*14-1*(-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*-2-14*-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*(-2)-1*-1`

`D=-4+1`

`D=-3`

Cramer's rule says that,

`x=(D_x)/(D)`

`\Rightarrow x=(18)/(-3)`

`\Rightarrow x=-6`

and

`y=(D_y)/(D)`

`\Rightarrow y=(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