Linear Algebra Examples

$[\begin{array}{c} 7 x & - 8 \\ 8 y & - 3 \end{array}] = [\begin{array}{c} 0 & 20 \\ 2 y & 3 \end{array}]$

Step 1

Find the function rule.

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Step 1.1

Check if the function rule is linear.

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Step 1.1.1

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 1.1.2

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

$20 = a (- 8) + b 3 = a (- 3) + b$

Step 1.1.3

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

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Step 1.1.3.1

Solve for $b$ in $20 = a (- 8) + b$ .

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Step 1.1.3.1.1

Rewrite the equation as $a (- 8) + b = 20$ .

$a (- 8) + b = 20$

$3 = a (- 3) + b$

Step 1.1.3.1.2

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

$- 8 a + b = 20$

$3 = a (- 3) + b$

Step 1.1.3.1.3

Add $8 a$ to both sides of the equation.

$b = 20 + 8 a$

$3 = a (- 3) + b$

$b = 20 + 8 a$

$3 = a (- 3) + b$

Step 1.1.3.2

Replace all occurrences of $b$ with $20 + 8 a$ in each equation.

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Step 1.1.3.2.1

Replace all occurrences of $b$ in $3 = a (- 3) + b$ with $20 + 8 a$ .

$3 = a (- 3) + 20 + 8 a$

$b = 20 + 8 a$

Step 1.1.3.2.2

Simplify $3 = a (- 3) + 20 + 8 a$ .

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Step 1.1.3.2.2.1

Simplify the left side.

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Step 1.1.3.2.2.1.1

Remove parentheses.

$3 = a (- 3) + 20 + 8 a$

$b = 20 + 8 a$

$3 = a (- 3) + 20 + 8 a$

$b = 20 + 8 a$

Step 1.1.3.2.2.2

Simplify the right side.

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Step 1.1.3.2.2.2.1

Simplify $a (- 3) + 20 + 8 a$ .

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Step 1.1.3.2.2.2.1.1

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

$3 = - 3 a + 20 + 8 a$

$b = 20 + 8 a$

Step 1.1.3.2.2.2.1.2

Add $- 3 a$ and $8 a$ .

$3 = 5 a + 20$

$b = 20 + 8 a$

$3 = 5 a + 20$

$b = 20 + 8 a$

$3 = 5 a + 20$

$b = 20 + 8 a$

$3 = 5 a + 20$

$b = 20 + 8 a$

$3 = 5 a + 20$

$b = 20 + 8 a$

Step 1.1.3.3

Solve for $a$ in $3 = 5 a + 20$ .

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Step 1.1.3.3.1

Rewrite the equation as $5 a + 20 = 3$ .

$5 a + 20 = 3$

$b = 20 + 8 a$

Step 1.1.3.3.2

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

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Step 1.1.3.3.2.1

Subtract $20$ from both sides of the equation.

$5 a = 3 - 20$

$b = 20 + 8 a$

Step 1.1.3.3.2.2

Subtract $20$ from $3$ .

$5 a = - 17$

$b = 20 + 8 a$

$5 a = - 17$

$b = 20 + 8 a$

Step 1.1.3.3.3

Divide each term in $5 a = - 17$ by $5$ and simplify.

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Step 1.1.3.3.3.1

Divide each term in $5 a = - 17$ by $5$ .

$\frac{5 a}{5} = \frac{- 17}{5}$

$b = 20 + 8 a$

Step 1.1.3.3.3.2

Simplify the left side.

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Step 1.1.3.3.3.2.1

Cancel the common factor of $5$ .

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Step 1.1.3.3.3.2.1.1

Cancel the common factor.

$\frac{5 a}{5} = \frac{- 17}{5}$

$b = 20 + 8 a$

Step 1.1.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 17}{5}$

$b = 20 + 8 a$

$a = \frac{- 17}{5}$

$b = 20 + 8 a$

$a = \frac{- 17}{5}$

$b = 20 + 8 a$

Step 1.1.3.3.3.3

Simplify the right side.

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Step 1.1.3.3.3.3.1

Move the negative in front of the fraction.

$a = - \frac{17}{5}$

$b = 20 + 8 a$

$a = - \frac{17}{5}$

$b = 20 + 8 a$

$a = - \frac{17}{5}$

$b = 20 + 8 a$

$a = - \frac{17}{5}$

$b = 20 + 8 a$

Step 1.1.3.4

Replace all occurrences of $a$ with $- \frac{17}{5}$ in each equation.

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Step 1.1.3.4.1

Replace all occurrences of $a$ in $b = 20 + 8 a$ with $- \frac{17}{5}$ .

$b = 20 + 8 (- \frac{17}{5})$

$a = - \frac{17}{5}$

Step 1.1.3.4.2

Simplify the right side.

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Step 1.1.3.4.2.1

Simplify $20 + 8 (- \frac{17}{5})$ .

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Step 1.1.3.4.2.1.1

Simplify each term.

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Step 1.1.3.4.2.1.1.1

Multiply $8 (- \frac{17}{5})$ .

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Step 1.1.3.4.2.1.1.1.1

Multiply $- 1$ by $8$ .

$b = 20 - 8 (\frac{17}{5})$

$a = - \frac{17}{5}$

Step 1.1.3.4.2.1.1.1.2

Combine $- 8$ and $\frac{17}{5}$ .

$b = 20 + \frac{- 8 \cdot 17}{5}$

$a = - \frac{17}{5}$

Step 1.1.3.4.2.1.1.1.3

Multiply $- 8$ by $17$ .

$b = 20 + \frac{- 136}{5}$

$a = - \frac{17}{5}$

$b = 20 + \frac{- 136}{5}$

$a = - \frac{17}{5}$

Step 1.1.3.4.2.1.1.2

Move the negative in front of the fraction.

$b = 20 - \frac{136}{5}$

$a = - \frac{17}{5}$

$b = 20 - \frac{136}{5}$

$a = - \frac{17}{5}$

Step 1.1.3.4.2.1.2

To write $20$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$b = 20 \cdot \frac{5}{5} - \frac{136}{5}$

$a = - \frac{17}{5}$

Step 1.1.3.4.2.1.3

Combine $20$ and $\frac{5}{5}$ .

$b = \frac{20 \cdot 5}{5} - \frac{136}{5}$

$a = - \frac{17}{5}$

Step 1.1.3.4.2.1.4

Combine the numerators over the common denominator.

$b = \frac{20 \cdot 5 - 136}{5}$

$a = - \frac{17}{5}$

Step 1.1.3.4.2.1.5

Simplify the numerator.

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Step 1.1.3.4.2.1.5.1

Multiply $20$ by $5$ .

$b = \frac{100 - 136}{5}$

$a = - \frac{17}{5}$

Step 1.1.3.4.2.1.5.2

Subtract $136$ from $100$ .

$b = \frac{- 36}{5}$

$a = - \frac{17}{5}$

$b = \frac{- 36}{5}$

$a = - \frac{17}{5}$

Step 1.1.3.4.2.1.6

Move the negative in front of the fraction.

$b = - \frac{36}{5}$

$a = - \frac{17}{5}$

$b = - \frac{36}{5}$

$a = - \frac{17}{5}$

$b = - \frac{36}{5}$

$a = - \frac{17}{5}$

$b = - \frac{36}{5}$

$a = - \frac{17}{5}$

Step 1.1.3.5

List all of the solutions.

$b = - \frac{36}{5}, a = - \frac{17}{5}$

Step 1.1.4

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 1.1.4.1

Calculate the value of $y$ when $a = - \frac{17}{5}$ , $b = - \frac{36}{5}$ , and $x = - 8$ .

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Step 1.1.4.1.1

Multiply $(- \frac{17}{5}) (- 8)$ .

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Step 1.1.4.1.1.1

Multiply $- 8$ by $- 1$ .

$y = 8 (\frac{17}{5}) - \frac{36}{5}$

Step 1.1.4.1.1.2

Combine $8$ and $\frac{17}{5}$ .

$y = \frac{8 \cdot 17}{5} - \frac{36}{5}$

Step 1.1.4.1.1.3

Multiply $8$ by $17$ .

$y = \frac{136}{5} - \frac{36}{5}$

Step 1.1.4.1.2

Combine fractions.

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Step 1.1.4.1.2.1

Combine the numerators over the common denominator.

$y = \frac{136 - 36}{5}$

Step 1.1.4.1.2.2

Simplify the expression.

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Step 1.1.4.1.2.2.1

Subtract $36$ from $136$ .

$y = \frac{100}{5}$

Step 1.1.4.1.2.2.2

Divide $100$ by $5$ .

$y = 20$

Step 1.1.4.2

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = - 8$ . This check passes since $y = 20$ and $y = 20$ .

$20 = 20$

Step 1.1.4.3

Calculate the value of $y$ when $a = - \frac{17}{5}$ , $b = - \frac{36}{5}$ , and $x = - 3$ .

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Step 1.1.4.3.1

Multiply $(- \frac{17}{5}) (- 3)$ .

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Step 1.1.4.3.1.1

Multiply $- 3$ by $- 1$ .

$y = 3 (\frac{17}{5}) - \frac{36}{5}$

Step 1.1.4.3.1.2

Combine $3$ and $\frac{17}{5}$ .

$y = \frac{3 \cdot 17}{5} - \frac{36}{5}$

Step 1.1.4.3.1.3

Multiply $3$ by $17$ .

$y = \frac{51}{5} - \frac{36}{5}$

Step 1.1.4.3.2

Combine fractions.

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Step 1.1.4.3.2.1

Combine the numerators over the common denominator.

$y = \frac{51 - 36}{5}$

Step 1.1.4.3.2.2

Simplify the expression.

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Step 1.1.4.3.2.2.1

Subtract $36$ from $51$ .

$y = \frac{15}{5}$

Step 1.1.4.3.2.2.2

Divide $15$ by $5$ .

$y = 3$

Step 1.1.4.4

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

$3 = 3$

Step 1.1.4.5

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

The function is linear

Step 1.2

Since all $y = y$ , the function is linear and follows the form $y = - \frac{17 x}{5} - \frac{36}{5}$ .

$y = - \frac{17 x}{5} - \frac{36}{5}$

Step 2

Find $7 x$ .

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Step 2.1

Use the function rule equation to find $7 x$ .

$0 = - \frac{177 x}{5} - \frac{36}{5}$

Step 2.2

Rewrite the equation as $- \frac{177 x}{5} - \frac{36}{5} = 0$ .

$- \frac{177 x}{5} - \frac{36}{5} = 0$

Step 2.3

Add $\frac{36}{5}$ to both sides of the equation.

$- \frac{177 x}{5} = \frac{36}{5}$

Step 2.4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- 177 x = 36$

Step 2.5

Divide each term in $- 177 x = 36$ by $- 177$ and simplify.

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Step 2.5.1

Divide each term in $- 177 x = 36$ by $- 177$ .

$\frac{- 177 x}{- 177} = \frac{36}{- 177}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $- 177$ .

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Step 2.5.2.1.1

Cancel the common factor.

$\frac{- 177 x}{- 177} = \frac{36}{- 177}$

Step 2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{36}{- 177}$

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Cancel the common factor of $36$ and $- 177$ .

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Step 2.5.3.1.1

Factor $3$ out of $36$ .

$x = \frac{3 (12)}{- 177}$

Step 2.5.3.1.2

Cancel the common factors.

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Step 2.5.3.1.2.1

Factor $3$ out of $- 177$ .

$x = \frac{3 \cdot 12}{3 \cdot - 59}$

Step 2.5.3.1.2.2

Cancel the common factor.

$x = \frac{3 \cdot 12}{3 \cdot - 59}$

Step 2.5.3.1.2.3

Rewrite the expression.

$x = \frac{12}{- 59}$

Step 2.5.3.2

Move the negative in front of the fraction.

$x = - \frac{12}{59}$

Step 3

Find $8 y$ .

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Step 3.1

Use the function rule equation to find $8 y$ .

$2 y = - \frac{178 y}{5} - \frac{36}{5}$

Step 3.2

Move all terms containing $y$ to the left side of the equation.

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Step 3.2.1

Add $\frac{178 y}{5}$ to both sides of the equation.

$2 y + \frac{178 y}{5} = - \frac{36}{5}$

Step 3.2.2

To write $2 y$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$2 y \cdot \frac{5}{5} + \frac{178 y}{5} = - \frac{36}{5}$

Step 3.2.3

Combine $2 y$ and $\frac{5}{5}$ .

$\frac{2 y \cdot 5}{5} + \frac{178 y}{5} = - \frac{36}{5}$

Step 3.2.4

Combine the numerators over the common denominator.

$\frac{2 y \cdot 5 + 178 y}{5} = - \frac{36}{5}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Factor $2 y$ out of $2 y \cdot 5 + 178 y$ .

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Step 3.2.5.1.1

Factor $2 y$ out of $2 y \cdot 5$ .

$\frac{2 y (5) + 178 y}{5} = - \frac{36}{5}$

Step 3.2.5.1.2

Factor $2 y$ out of $178 y$ .

$\frac{2 y (5) + 2 y (89)}{5} = - \frac{36}{5}$

Step 3.2.5.1.3

Factor $2 y$ out of $2 y (5) + 2 y (89)$ .

$\frac{2 y (5 + 89)}{5} = - \frac{36}{5}$

Step 3.2.5.2

Add $5$ and $89$ .

$\frac{2 y \cdot 94}{5} = - \frac{36}{5}$

Step 3.2.5.3

Multiply $94$ by $2$ .

$\frac{188 y}{5} = - \frac{36}{5}$

Step 3.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$188 y = - 36$

Step 3.4

Divide each term in $188 y = - 36$ by $188$ and simplify.

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Step 3.4.1

Divide each term in $188 y = - 36$ by $188$ .

$\frac{188 y}{188} = \frac{- 36}{188}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $188$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{188 y}{188} = \frac{- 36}{188}$

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 36}{188}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Cancel the common factor of $- 36$ and $188$ .

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Step 3.4.3.1.1

Factor $4$ out of $- 36$ .

$y = \frac{4 (- 9)}{188}$

Step 3.4.3.1.2

Cancel the common factors.

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Step 3.4.3.1.2.1

Factor $4$ out of $188$ .

$y = \frac{4 \cdot - 9}{4 \cdot 47}$

Step 3.4.3.1.2.2

Cancel the common factor.

$y = \frac{4 \cdot - 9}{4 \cdot 47}$

Step 3.4.3.1.2.3

Rewrite the expression.

$y = \frac{- 9}{47}$

Step 3.4.3.2

Move the negative in front of the fraction.

$y = - \frac{9}{47}$

Step 4

List all of the solutions.

$x = - \frac{12}{59} y = - \frac{9}{47}$