Solve the Matrix Equation [[1,2],[3,7]][[x],[y]]=[[10],[21]]

$[\begin{array}{c} 1 & 2 \\ 3 & 7 \end{array}] [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 10 \\ 21 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 1 & 2 \\ 3 & 7 \end{array}] [\begin{array}{c} x \\ y \end{array}]$ .

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} 1 x + 2 y \\ 3 x + 7 y \end{array}] = [\begin{array}{c} 10 \\ 21 \end{array}]$

Simplify each element of the matrix by multiplying out all the expressions.

$[\begin{array}{c} x + 2 y \\ 3 x + 7 y \end{array}] = [\begin{array}{c} 10 \\ 21 \end{array}]$

Step 2

Write as a linear system of equations.

$x + 2 y = 10 3 x + 7 y = 21$

Step 3

Solve the system of equations.

Subtract $2 y$ from both sides of the equation.

$x = 10 - 2 y$

$3 x + 7 y = 21$

Replace all occurrences of $x$ with $10 - 2 y$ in each equation.

Replace all occurrences of $x$ in $3 x + 7 y = 21$ with $10 - 2 y$ .

$3 (10 - 2 y) + 7 y = 21$

$x = 10 - 2 y$

Simplify the left side.

Simplify $3 (10 - 2 y) + 7 y$ .

Simplify each term.

Apply the distributive property.

$3 \cdot 10 + 3 (- 2 y) + 7 y = 21$

$x = 10 - 2 y$

Multiply $3$ by $10$ .

$30 + 3 (- 2 y) + 7 y = 21$

$x = 10 - 2 y$

Multiply $- 2$ by $3$ .

$30 - 6 y + 7 y = 21$

$x = 10 - 2 y$

$30 - 6 y + 7 y = 21$

$x = 10 - 2 y$

Add $- 6 y$ and $7 y$ .

$30 + y = 21$

$x = 10 - 2 y$

$30 + y = 21$

$x = 10 - 2 y$

$30 + y = 21$

$x = 10 - 2 y$

$30 + y = 21$

$x = 10 - 2 y$

Move all terms not containing $y$ to the right side of the equation.

Subtract $30$ from both sides of the equation.

$y = 21 - 30$

$x = 10 - 2 y$

Subtract $30$ from $21$ .

$y = - 9$

$x = 10 - 2 y$

$y = - 9$

$x = 10 - 2 y$

Replace all occurrences of $y$ with $- 9$ in each equation.

Replace all occurrences of $y$ in $x = 10 - 2 y$ with $- 9$ .

$x = 10 - 2 \cdot - 9$

$y = - 9$

Simplify the right side.

Simplify $10 - 2 \cdot - 9$ .

Multiply $- 2$ by $- 9$ .

$x = 10 + 18$

$y = - 9$

Add $10$ and $18$ .

$x = 28$

$y = - 9$

$x = 28$

$y = - 9$

$x = 28$

$y = - 9$

$x = 28$

$y = - 9$

List all of the solutions.

$x = 28, y = - 9$

Solve the Matrix Equation [[1,2],[3,7]][[x],[y]]=[[10],[21]]

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