Question

Solve the system by using a matrix equation (Picture provided)

Answer 1

Option b is correct (8,13).

Explanation:

7x - 4y = 4

10x - 6y =2

it can be represented in matrix form as
`\left[\begin{array}{cc}7&-4\\10&-6\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] = \left[\begin{array}{c}4\\2\end{array}\right]`

A=
`\left[\begin{array}{cc}7&-4\\10&-6\end{array}\right]`

X=
`\left[\begin{array}{c}x\\y\end{array}\right]`

B=
`\left[\begin{array}{c}4\\2\end{array}\right]`

i.e, AX=B

or X= A⁻¹ B

A⁻¹ = 1/|A| * Adj A

determinant of A = |A|= (7*-6) - (-4*10)

= (-42)-(-40)

= (-42) + 40 = -2

so, |A| = -2

Adj A=
`\left[\begin{array}{cc}-6&4\\-10&7\end{array}\right]`

A⁻¹ =
`\left[\begin{array}{cc}-6&4\\-10&7\end{array}\right]`/ -2

A⁻¹ =
`\left[\begin{array}{cc}3&-2\\5&-7/2\end{array}\right]`

X= A⁻¹ B

X=
`\left[\begin{array}{cc}3&-2\\5&-7/2\end{array}\right] *\left[\begin{array}{c}4\\2\end{array}\right]`

X=
`\left[\begin{array}{c}(3*4) + (-2*2)\\(5*4) + (-7/2*2)\end{array}\right]`

X=
`\left[\begin{array}{c}12-4\\20-7\end{array}\right]`

X=
`\left[\begin{array}{c}8\\13\end{array}\right]`

x= 8, y= 13

solution set= (8,13).

Option b is correct.