Question

Solving a Two-Step Matrix Equation
Solve the equation:

Answer 1

`\boxed {x_(1) = 3}`

`\boxed {x_(2) = -4}`

Explanation:

Solve the following equation:

`\left[\begin{array}{ccc}3&2\\5&5\\\end{array}\right] \left[\begin{array}{ccc}x_(1)\\x_(2)\\\end{array}\right] + \left[\begin{array}{ccc}1\\2\\\end{array}\right] = \left[\begin{array}{ccc}2\\-3\\\end{array}\right]`

-In order to solve a pair of equations by using substitution, you first need to solve one of the equations for one of variables and then you would substitute the result for that variable in the other equation:

-First equation:

`3x_(1) + 2x_(2) + 1 = 2`

-Second equation:

`5x_(1) + 5x_(2) + 2 = -3`

-Choose one of the two following equations, which I choose the first one, then you solve for
`x_(1)` by isolating

`3x_(1) + 2x_(2) + 1 = 2`

-Subtract
`1` to both sides:

`3x_(1) + 2x_(2) + 1 - 1 = 2 - 1`

`3x_(1) + 2x_(2) = 1`

-Subtract
`2x_(2)` to both sides:

`3x_(1) + 2x_(2) - 2x_(2) = -2x_(2) + 1`

`3x_(1) = -2x_(2) + 1`

-Divide both sides by
`3`:

`3x_(1) = -2x_(2) + 1`

`x_(1) = (1)/(3) (-2x_(2) + 1)`

-Multiply
`-2x_(2) + 1` by
`(1)/(3)`:

`x_(1) = (1)/(3) (-2x_(2) + 1)`

`x_(1) = -(2)/(3)x_(2) + (1)/(3)`

-Substitute
`-(2x_(2) + 1)/(3)` for
`x_(1)` in the second equation, which is
`5x_(1) + 5x_(2) + 2 = -3`:

`5x_(1) + 5x_(2) + 2 = -3`

`5(-(2)/(3)x_(2) + (1)/(3)) + 5x_(2) + 2 = -3`

Multiply
`-(2x_(2) + 1)/(3)` by
`5`:

`5(-(2)/(3)x_(2) + (1)/(3)) + 5x_(2) + 2 = -3`

`-(10)/(3)x_(2) + (5)/(3) + 5x_(2) + 2 = -3`

-Combine like terms:

`-(10)/(3)x_(2) + (5)/(3) + 5x_(2) + 2 = -3`

`(5)/(3)x_(2) + (11)/(3) = -3`

-Subtract
`(11)/(3)` to both sides:

`(5)/(3)x_(2) + (11)/(3) - (11)/(3) = -3 - (11)/(3)`

`(5)/(3)x_(2) = -(20)/(3)`

-Multiply both sides by
`(5)/(3)`:

`((5)/(3)x_(2))/((5)/(3)) = (-(20)/(3))/((5)/(3))`

`\boxed {x_(2) = -4}`

-After you have the value of
`x_2`, substitute for
`x_(2)` onto this equation, which is
`x_(1) = -(2)/(3)x_(2) + (1)/(3)`:

`x_(1) = -(2)/(3)x_(2) + (1)/(3)`

`x_(1) = -(2)/(3)(-4) + (1)/(3)`

-Multiply
`-(2)/(3)` and
`-4`:

`x_(1) = -(2)/(3)(-4) + (1)/(3)`

`x_(1) = (8 + 1)/(3)`

-Since both
`(1)/(3)` and
`(8)/(3)` have the same denominator, then add the numerators together. Also, after you have added both numerators together, reduce the fraction to the lowest term:

`x_(1) = (8 + 1)/(3)`

`x_(1) = (9)/(3)`

`\boxed {x_(1) = 3}`