Algebra Examples

$[\begin{array}{c} 2 x & - y \\ - 4 & 3 \end{array}] \cdot [\begin{array}{c} 1 & 6 \\ - 2 & 5 \end{array}] = [\begin{array}{c} 6 & - 134 \\ - 10 & - 9 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 2 x & - y \\ - 4 & 3 \end{array}] [\begin{array}{c} 1 & 6 \\ - 2 & 5 \end{array}]$ .

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Step 1.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $2 \times 2$ and the second matrix is $2 \times 2$ .

Step 1.2

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

$[\begin{array}{c} 2 x \cdot 1 - y \cdot - 2 & 2 x \cdot 6 - y \cdot 5 \\ - 4 \cdot 1 + 3 \cdot - 2 & - 4 \cdot 6 + 3 \cdot 5 \end{array}] = [\begin{array}{c} 6 & - 134 \\ - 10 & - 9 \end{array}]$

Step 1.3

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

$[\begin{array}{c} 2 x + 2 y & 12 x - 5 y \\ - 10 & - 9 \end{array}] = [\begin{array}{c} 6 & - 134 \\ - 10 & - 9 \end{array}]$

Step 2

Find the function rule.

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Step 2.1

Check if the function rule is linear.

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Step 2.1.1

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

$y = a x + b$

Step 2.1.2

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

$- 10 = a (- 10) + b - 9 = a (- 9) + b$

Step 2.1.3

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

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Step 2.1.3.1

Solve for $b$ in $- 10 = a (- 10) + b$ .

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Step 2.1.3.1.1

Rewrite the equation as $a (- 10) + b = - 10$ .

$a (- 10) + b = - 10$

$- 9 = a (- 9) + b$

Step 2.1.3.1.2

Move $- 10$ to the left of $a$ .

$- 10 a + b = - 10$

$- 9 = a (- 9) + b$

Step 2.1.3.1.3

Add $10 a$ to both sides of the equation.

$b = - 10 + 10 a$

$- 9 = a (- 9) + b$

$b = - 10 + 10 a$

$- 9 = a (- 9) + b$

Step 2.1.3.2

Replace all occurrences of $b$ with $- 10 + 10 a$ in each equation.

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Step 2.1.3.2.1

Replace all occurrences of $b$ in $- 9 = a (- 9) + b$ with $- 10 + 10 a$ .

$- 9 = a (- 9) - 10 + 10 a$

$b = - 10 + 10 a$

Step 2.1.3.2.2

Simplify $- 9 = a (- 9) - 10 + 10 a$ .

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Step 2.1.3.2.2.1

Simplify the left side.

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Step 2.1.3.2.2.1.1

Remove parentheses.

$- 9 = a (- 9) - 10 + 10 a$

$b = - 10 + 10 a$

$- 9 = a (- 9) - 10 + 10 a$

$b = - 10 + 10 a$

Step 2.1.3.2.2.2

Simplify the right side.

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Step 2.1.3.2.2.2.1

Simplify $a (- 9) - 10 + 10 a$ .

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Step 2.1.3.2.2.2.1.1

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

$- 9 = - 9 a - 10 + 10 a$

$b = - 10 + 10 a$

Step 2.1.3.2.2.2.1.2

Add $- 9 a$ and $10 a$ .

$- 9 = a - 10$

$b = - 10 + 10 a$

$- 9 = a - 10$

$b = - 10 + 10 a$

$- 9 = a - 10$

$b = - 10 + 10 a$

$- 9 = a - 10$

$b = - 10 + 10 a$

$- 9 = a - 10$

$b = - 10 + 10 a$

Step 2.1.3.3

Solve for $a$ in $- 9 = a - 10$ .

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Step 2.1.3.3.1

Rewrite the equation as $a - 10 = - 9$ .

$a - 10 = - 9$

$b = - 10 + 10 a$

Step 2.1.3.3.2

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

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Step 2.1.3.3.2.1

Add $10$ to both sides of the equation.

$a = - 9 + 10$

$b = - 10 + 10 a$

Step 2.1.3.3.2.2

Add $- 9$ and $10$ .

$a = 1$

$b = - 10 + 10 a$

$a = 1$

$b = - 10 + 10 a$

$a = 1$

$b = - 10 + 10 a$

Step 2.1.3.4

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

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Step 2.1.3.4.1

Replace all occurrences of $a$ in $b = - 10 + 10 a$ with $1$ .

$b = - 10 + 10 (1)$

$a = 1$

Step 2.1.3.4.2

Simplify the right side.

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Step 2.1.3.4.2.1

Simplify $- 10 + 10 (1)$ .

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Step 2.1.3.4.2.1.1

Multiply $10$ by $1$ .

$b = - 10 + 10$

$a = 1$

Step 2.1.3.4.2.1.2

Add $- 10$ and $10$ .

$b = 0$

$a = 1$

$b = 0$

$a = 1$

$b = 0$

$a = 1$

$b = 0$

$a = 1$

Step 2.1.3.5

List all of the solutions.

$b = 0, a = 1$

Step 2.1.4

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

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Step 2.1.4.1

Calculate the value of $y$ when $a = 1$ , $b = 0$ , and $x = - 10$ .

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Step 2.1.4.1.1

Multiply $- 10$ by $1$ .

$y = - 10 + 0$

Step 2.1.4.1.2

Add $- 10$ and $0$ .

$y = - 10$

Step 2.1.4.2

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = - 10$ . This check passes since $y = - 10$ and $y = - 10$ .

$- 10 = - 10$

Step 2.1.4.3

Calculate the value of $y$ when $a = 1$ , $b = 0$ , and $x = - 9$ .

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Step 2.1.4.3.1

Multiply $- 9$ by $1$ .

$y = - 9 + 0$

Step 2.1.4.3.2

Add $- 9$ and $0$ .

$y = - 9$

Step 2.1.4.4

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

$- 9 = - 9$

Step 2.1.4.5

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

The function is linear

Step 2.2

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

$y = x$

Step 3

Find $2 x + 2 y$ .

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Step 3.1

Use the function rule equation to find $2 x + 2 y$ .

$6 = 2 x + 2 y$

Step 3.2

Rewrite the equation as $2 x + 2 y = 6$ .

$2 x + 2 y = 6$

Step 3.3

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

$2 x = 6 - 2 y$

Step 3.4

Divide each term in $2 x = 6 - 2 y$ by $2$ and simplify.

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Step 3.4.1

Divide each term in $2 x = 6 - 2 y$ by $2$ .

$\frac{2 x}{2} = \frac{6}{2} + \frac{- 2 y}{2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{6}{2} + \frac{- 2 y}{2}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{2} + \frac{- 2 y}{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify each term.

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Step 3.4.3.1.1

Divide $6$ by $2$ .

$x = 3 + \frac{- 2 y}{2}$

Step 3.4.3.1.2

Cancel the common factor of $- 2$ and $2$ .

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Step 3.4.3.1.2.1

Factor $2$ out of $- 2 y$ .

$x = 3 + \frac{2 (- y)}{2}$

Step 3.4.3.1.2.2

Cancel the common factors.

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Step 3.4.3.1.2.2.1

Factor $2$ out of $2$ .

$x = 3 + \frac{2 (- y)}{2 (1)}$

Step 3.4.3.1.2.2.2

Cancel the common factor.

$x = 3 + \frac{2 (- y)}{2 \cdot 1}$

Step 3.4.3.1.2.2.3

Rewrite the expression.

$x = 3 + \frac{- y}{1}$

Step 3.4.3.1.2.2.4

Divide $- y$ by $1$ .

$x = 3 - y$

Step 4

Find $12 x - 5 y$ .

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Step 4.1

Use the function rule equation to find $12 x - 5 y$ .

$- 134 = 12 x - 5 y$

Step 4.2

Rewrite the equation as $12 x - 5 y = - 134$ .

$12 x - 5 y = - 134$

Step 4.3

Add $5 y$ to both sides of the equation.

$12 x = - 134 + 5 y$

Step 4.4

Divide each term in $12 x = - 134 + 5 y$ by $12$ and simplify.

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Step 4.4.1

Divide each term in $12 x = - 134 + 5 y$ by $12$ .

$\frac{12 x}{12} = \frac{- 134}{12} + \frac{5 y}{12}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $12$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{12 x}{12} = \frac{- 134}{12} + \frac{5 y}{12}$

Step 4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 134}{12} + \frac{5 y}{12}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Simplify each term.

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Step 4.4.3.1.1

Cancel the common factor of $- 134$ and $12$ .

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Step 4.4.3.1.1.1

Factor $2$ out of $- 134$ .

$x = \frac{2 (- 67)}{12} + \frac{5 y}{12}$

Step 4.4.3.1.1.2

Cancel the common factors.

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Step 4.4.3.1.1.2.1

Factor $2$ out of $12$ .

$x = \frac{2 \cdot - 67}{2 \cdot 6} + \frac{5 y}{12}$

Step 4.4.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 67}{2 \cdot 6} + \frac{5 y}{12}$

Step 4.4.3.1.1.2.3

Rewrite the expression.

$x = \frac{- 67}{6} + \frac{5 y}{12}$

Step 4.4.3.1.2

Move the negative in front of the fraction.

$x = - \frac{67}{6} + \frac{5 y}{12}$

Step 5

List all of the solutions.

$x = 3 - y x = - \frac{67}{6} + \frac{5 y}{12}$