Solve the Matrix Equation [[4,2],[0,1]]+[[5,9],[8,2]]=[[9,x],[8,3]]

$[\begin{array}{c} 4 & 2 \\ 0 & 1 \end{array}] + [\begin{array}{c} 5 & 9 \\ 8 & 2 \end{array}] = [\begin{array}{c} 9 & x \\ 8 & 3 \end{array}]$

Step 1

Simplify the right side of the equation.

Add the corresponding elements.

$X = [\begin{array}{c} 4 + 5 & 2 + 9 \\ 0 + 8 & 1 + 2 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} 4 + 5 & 2 + 9 \\ 0 + 8 & 1 + 2 \end{array}]$ .

Simplify $4 + 5$ .

$X = [\begin{array}{c} 9 & 2 + 9 \\ 0 + 8 & 1 + 2 \end{array}]$

Simplify $2 + 9$ .

$X = [\begin{array}{c} 9 & 11 \\ 0 + 8 & 1 + 2 \end{array}]$

Simplify $0 + 8$ .

$X = [\begin{array}{c} 9 & 11 \\ 8 & 1 + 2 \end{array}]$

Simplify $1 + 2$ .

$X = [\begin{array}{c} 9 & 11 \\ 8 & 3 \end{array}]$

Step 2

Solve.

$9 = 9, x = 11, 8 = 8, 3 = 3$

Step 3

Find the function rule.

Check if the function rule is linear.

To find if the table follows a function rule, check to see if the values follow the linear form $y = a x + b$ .

$y = a x + b$

Build a set of equations from the table such that $y = a x + b$ .

$9 = a (4 + 5) + b 8 = a (0 + 8) + b 3 = a (1 + 2) + b$

Calculate the values of $a$ and $b$ .

Solve for $b$ in $9 = a \cdot 9 + b$ .

Rewrite the equation as $a \cdot 9 + b = 9$ .

$a \cdot 9 + b = 9$

$8 = a \cdot 8 + b$

$3 = a \cdot 3 + b$

Move $9$ to the left of $a$ .

$9 a + b = 9$

$8 = a \cdot 8 + b$

$3 = a \cdot 3 + b$

Subtract $9 a$ from both sides of the equation.

$b = 9 - 9 a$

$8 = a \cdot 8 + b$

$3 = a \cdot 3 + b$

$b = 9 - 9 a$

$8 = a \cdot 8 + b$

$3 = a \cdot 3 + b$

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

Replace all occurrences of $b$ in $8 = a \cdot 8 + b$ with $9 - 9 a$ .

$8 = a \cdot 8 + 9 - 9 a$

$b = 9 - 9 a$

$3 = a \cdot 3 + b$

Simplify $8 = a \cdot 8 + 9 - 9 a$ .

Simplify the left side.

Remove parentheses.

$8 = a \cdot 8 + 9 - 9 a$

$b = 9 - 9 a$

$3 = a \cdot 3 + b$

$8 = a \cdot 8 + 9 - 9 a$

$b = 9 - 9 a$

$3 = a \cdot 3 + b$

Simplify the right side.

Simplify $a \cdot 8 + 9 - 9 a$ .

Move $8$ to the left of $a$ .

$8 = 8 a + 9 - 9 a$

$b = 9 - 9 a$

$3 = a \cdot 3 + b$

Subtract $9 a$ from $8 a$ .

$8 = - a + 9$

$b = 9 - 9 a$

$3 = a \cdot 3 + b$

$8 = - a + 9$

$b = 9 - 9 a$

$3 = a \cdot 3 + b$

$8 = - a + 9$

$b = 9 - 9 a$

$3 = a \cdot 3 + b$

$8 = - a + 9$

$b = 9 - 9 a$

$3 = a \cdot 3 + b$

Replace all occurrences of $b$ in $3 = a \cdot 3 + b$ with $9 - 9 a$ .

$3 = a \cdot 3 + 9 - 9 a$

$8 = - a + 9$

$b = 9 - 9 a$

Simplify $3 = a \cdot 3 + 9 - 9 a$ .

Simplify the left side.

Remove parentheses.

$3 = a \cdot 3 + 9 - 9 a$

$8 = - a + 9$

$b = 9 - 9 a$

$3 = a \cdot 3 + 9 - 9 a$

$8 = - a + 9$

$b = 9 - 9 a$

Simplify the right side.

Simplify $a \cdot 3 + 9 - 9 a$ .

Move $3$ to the left of $a$ .

$3 = 3 a + 9 - 9 a$

$8 = - a + 9$

$b = 9 - 9 a$

Subtract $9 a$ from $3 a$ .

$3 = - 6 a + 9$

$8 = - a + 9$

$b = 9 - 9 a$

$3 = - 6 a + 9$

$8 = - a + 9$

$b = 9 - 9 a$

$3 = - 6 a + 9$

$8 = - a + 9$

$b = 9 - 9 a$

$3 = - 6 a + 9$

$8 = - a + 9$

$b = 9 - 9 a$

$3 = - 6 a + 9$

$8 = - a + 9$

$b = 9 - 9 a$

Solve for $a$ in $3 = - 6 a + 9$ .

Rewrite the equation as $- 6 a + 9 = 3$ .

$- 6 a + 9 = 3$

$8 = - a + 9$

$b = 9 - 9 a$

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

Subtract $9$ from both sides of the equation.

$- 6 a = 3 - 9$

$8 = - a + 9$

$b = 9 - 9 a$

Subtract $9$ from $3$ .

$- 6 a = - 6$

$8 = - a + 9$

$b = 9 - 9 a$

$- 6 a = - 6$

$8 = - a + 9$

$b = 9 - 9 a$

Divide each term in $- 6 a = - 6$ by $- 6$ and simplify.

Divide each term in $- 6 a = - 6$ by $- 6$ .

$\frac{- 6 a}{- 6} = \frac{- 6}{- 6}$

$8 = - a + 9$

$b = 9 - 9 a$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 a}{- 6} = \frac{- 6}{- 6}$

$8 = - a + 9$

$b = 9 - 9 a$

Divide $a$ by $1$ .

$a = \frac{- 6}{- 6}$

$8 = - a + 9$

$b = 9 - 9 a$

$a = \frac{- 6}{- 6}$

$8 = - a + 9$

$b = 9 - 9 a$

$a = \frac{- 6}{- 6}$

$8 = - a + 9$

$b = 9 - 9 a$

Simplify the right side.

Divide $- 6$ by $- 6$ .

$a = 1$

$8 = - a + 9$

$b = 9 - 9 a$

$a = 1$

$8 = - a + 9$

$b = 9 - 9 a$

$a = 1$

$8 = - a + 9$

$b = 9 - 9 a$

$a = 1$

$8 = - a + 9$

$b = 9 - 9 a$

Replace all occurrences of $a$ with $1$ in each equation.

Replace all occurrences of $a$ in $8 = - a + 9$ with $1$ .

$8 = - (1) + 9$

$a = 1$

$b = 9 - 9 a$

Simplify the right side.

Simplify $- (1) + 9$ .

Multiply $- 1$ by $1$ .

$8 = - 1 + 9$

$a = 1$

$b = 9 - 9 a$

Add $- 1$ and $9$ .

$8 = 8$

$a = 1$

$b = 9 - 9 a$

$8 = 8$

$a = 1$

$b = 9 - 9 a$

$8 = 8$

$a = 1$

$b = 9 - 9 a$

Replace all occurrences of $a$ in $b = 9 - 9 a$ with $1$ .

$b = 9 - 9 \cdot 1$

$8 = 8$

$a = 1$

Simplify the right side.

Simplify $9 - 9 \cdot 1$ .

Multiply $- 9$ by $1$ .

$b = 9 - 9$

$8 = 8$

$a = 1$

Subtract $9$ from $9$ .

$b = 0$

$8 = 8$

$a = 1$

$b = 0$

$8 = 8$

$a = 1$

$b = 0$

$8 = 8$

$a = 1$

$b = 0$

$8 = 8$

$a = 1$

Remove any equations from the system that are always true.

$b = 0$

$a = 1$

List all of the solutions.

$b = 0, a = 1$

Calculate the value of $y$ using each $x$ value in the relation and compare this value to the given $y$ value in the relation.

Calculate the value of $y$ when $a = 1$ , $b = 0$ , and $x = 4 + 5$ .

Simplify each term.

Multiply $4 + 5$ by $1$ .

$y = 4 + 5 + 0$

Add $4$ and $5$ .

$y = 9 + 0$

Add $9$ and $0$ .

$y = 9$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 4 + 5$ . This check passes since $y = 9$ and $y = 9$ .

$9 = 9$

Calculate the value of $y$ when $a = 1$ , $b = 0$ , and $x = 0 + 8$ .

Simplify each term.

Multiply $0 + 8$ by $1$ .

$y = 0 + 8 + 0$

Add $0$ and $8$ .

$y = 8 + 0$

Add $8$ and $0$ .

$y = 8$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 0 + 8$ . This check passes since $y = 8$ and $y = 8$ .

$8 = 8$

Calculate the value of $y$ when $a = 1$ , $b = 0$ , and $x = 1 + 2$ .

Simplify each term.

Multiply $1 + 2$ by $1$ .

$y = 1 + 2 + 0$

Add $1$ and $2$ .

$y = 3 + 0$

Add $3$ and $0$ .

$y = 3$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 1 + 2$ . This check passes since $y = 3$ and $y = 3$ .

$3 = 3$

Since $y = y$ for the corresponding $x$ values, the function is linear.

The function is linear

Since all $y = y$ , the function is linear and follows the form $y = x$ .

$y = x$

Step 4

Find $x$ .

Use the function rule equation to find $x$ .

$x = 2 + 9$

Simplify.

$x = 11$

Solve the Matrix Equation [[4,2],[0,1]]+[[5,9],[8,2]]=[[9,x],[8,3]]

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