Linear Algebra Examples

$[\begin{array}{c} t + 1 & - 8 & 4 \\ 5 & - 6 & - 3 \end{array}] = [\begin{array}{c} 3 & - 8 & 4 \\ 5 & x - 7 & - 3 \end{array}]$

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 $y = a x + b$ .

$- 8 = a (- 8) + b 4 = a (4) + b 5 = a (5) + b - 3 = a (- 3) + b$

Step 1.1.3

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

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Step 1.1.3.1

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

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Step 1.1.3.1.1

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

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

$4 = a (4) + b$

$5 = a (5) + b$

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

Step 1.1.3.1.2

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

$- 8 a + b = - 8$

$4 = a (4) + b$

$5 = a (5) + b$

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

Step 1.1.3.1.3

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

$b = - 8 + 8 a$

$4 = a (4) + b$

$5 = a (5) + b$

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

$b = - 8 + 8 a$

$4 = a (4) + b$

$5 = a (5) + b$

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

Step 1.1.3.2

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

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Step 1.1.3.2.1

Replace all occurrences of $b$ in $4 = a (4) + b$ with $- 8 + 8 a$ .

$4 = a (4) - 8 + 8 a$

$b = - 8 + 8 a$

$5 = a (5) + b$

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

Step 1.1.3.2.2

Simplify $4 = a (4) - 8 + 8 a$ .

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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.

$4 = a (4) - 8 + 8 a$

$b = - 8 + 8 a$

$5 = a (5) + b$

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

$4 = a (4) - 8 + 8 a$

$b = - 8 + 8 a$

$5 = a (5) + b$

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

Step 1.1.3.2.2.2

Simplify the right side.

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Step 1.1.3.2.2.2.1

Simplify $a (4) - 8 + 8 a$ .

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Step 1.1.3.2.2.2.1.1

Move $4$ to the left of $a$ .

$4 = 4 a - 8 + 8 a$

$b = - 8 + 8 a$

$5 = a (5) + b$

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

Step 1.1.3.2.2.2.1.2

Add $4 a$ and $8 a$ .

$4 = 12 a - 8$

$b = - 8 + 8 a$

$5 = a (5) + b$

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

$4 = 12 a - 8$

$b = - 8 + 8 a$

$5 = a (5) + b$

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

$4 = 12 a - 8$

$b = - 8 + 8 a$

$5 = a (5) + b$

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

$4 = 12 a - 8$

$b = - 8 + 8 a$

$5 = a (5) + b$

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

Step 1.1.3.2.3

Replace all occurrences of $b$ in $5 = a (5) + b$ with $- 8 + 8 a$ .

$5 = a (5) - 8 + 8 a$

$4 = 12 a - 8$

$b = - 8 + 8 a$

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

Step 1.1.3.2.4

Simplify $5 = a (5) - 8 + 8 a$ .

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Step 1.1.3.2.4.1

Simplify the left side.

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Step 1.1.3.2.4.1.1

Remove parentheses.

$5 = a (5) - 8 + 8 a$

$4 = 12 a - 8$

$b = - 8 + 8 a$

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

$5 = a (5) - 8 + 8 a$

$4 = 12 a - 8$

$b = - 8 + 8 a$

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

Step 1.1.3.2.4.2

Simplify the right side.

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Step 1.1.3.2.4.2.1

Simplify $a (5) - 8 + 8 a$ .

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Step 1.1.3.2.4.2.1.1

Move $5$ to the left of $a$ .

$5 = 5 a - 8 + 8 a$

$4 = 12 a - 8$

$b = - 8 + 8 a$

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

Step 1.1.3.2.4.2.1.2

Add $5 a$ and $8 a$ .

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

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

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

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

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

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

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

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

Step 1.1.3.2.5

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

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

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

Step 1.1.3.2.6

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

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Step 1.1.3.2.6.1

Simplify the left side.

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Step 1.1.3.2.6.1.1

Remove parentheses.

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

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

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

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

Step 1.1.3.2.6.2

Simplify the right side.

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Step 1.1.3.2.6.2.1

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

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Step 1.1.3.2.6.2.1.1

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

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

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

Step 1.1.3.2.6.2.1.2

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

$- 3 = 5 a - 8$

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

$- 3 = 5 a - 8$

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

$- 3 = 5 a - 8$

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

$- 3 = 5 a - 8$

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

$- 3 = 5 a - 8$

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

Step 1.1.3.3

Solve for $a$ in $- 3 = 5 a - 8$ .

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Step 1.1.3.3.1

Rewrite the equation as $5 a - 8 = - 3$ .

$5 a - 8 = - 3$

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

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

Add $8$ to both sides of the equation.

$5 a = - 3 + 8$

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

Step 1.1.3.3.2.2

Add $- 3$ and $8$ .

$5 a = 5$

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

$5 a = 5$

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

Step 1.1.3.3.3

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

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Step 1.1.3.3.3.1

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

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

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

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.

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

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

Step 1.1.3.3.3.2.1.2

Divide $a$ by $1$ .

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

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

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

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

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

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

Step 1.1.3.3.3.3

Simplify the right side.

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Step 1.1.3.3.3.3.1

Divide $5$ by $5$ .

$a = 1$

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

$a = 1$

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

$a = 1$

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

$a = 1$

$5 = 13 a - 8$

$4 = 12 a - 8$

$b = - 8 + 8 a$

Step 1.1.3.4

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

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Step 1.1.3.4.1

Replace all occurrences of $a$ in $5 = 13 a - 8$ with $1$ .

$5 = 13 (1) - 8$

$a = 1$

$4 = 12 a - 8$

$b = - 8 + 8 a$

Step 1.1.3.4.2

Simplify the right side.

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Step 1.1.3.4.2.1

Simplify $13 (1) - 8$ .

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Step 1.1.3.4.2.1.1

Multiply $13$ by $1$ .

$5 = 13 - 8$

$a = 1$

$4 = 12 a - 8$

$b = - 8 + 8 a$

Step 1.1.3.4.2.1.2

Subtract $8$ from $13$ .

$5 = 5$

$a = 1$

$4 = 12 a - 8$

$b = - 8 + 8 a$

$5 = 5$

$a = 1$

$4 = 12 a - 8$

$b = - 8 + 8 a$

$5 = 5$

$a = 1$

$4 = 12 a - 8$

$b = - 8 + 8 a$

Step 1.1.3.4.3

Replace all occurrences of $a$ in $4 = 12 a - 8$ with $1$ .

$4 = 12 (1) - 8$

$5 = 5$

$a = 1$

$b = - 8 + 8 a$

Step 1.1.3.4.4

Simplify the right side.

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Step 1.1.3.4.4.1

Simplify $12 (1) - 8$ .

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Step 1.1.3.4.4.1.1

Multiply $12$ by $1$ .

$4 = 12 - 8$

$5 = 5$

$a = 1$

$b = - 8 + 8 a$

Step 1.1.3.4.4.1.2

Subtract $8$ from $12$ .

$4 = 4$

$5 = 5$

$a = 1$

$b = - 8 + 8 a$

$4 = 4$

$5 = 5$

$a = 1$

$b = - 8 + 8 a$

$4 = 4$

$5 = 5$

$a = 1$

$b = - 8 + 8 a$

Step 1.1.3.4.5

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

$b = - 8 + 8 (1)$

$4 = 4$

$5 = 5$

$a = 1$

Step 1.1.3.4.6

Simplify the right side.

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Step 1.1.3.4.6.1

Simplify $- 8 + 8 (1)$ .

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Step 1.1.3.4.6.1.1

Multiply $8$ by $1$ .

$b = - 8 + 8$

$4 = 4$

$5 = 5$

$a = 1$

Step 1.1.3.4.6.1.2

Add $- 8$ and $8$ .

$b = 0$

$4 = 4$

$5 = 5$

$a = 1$

$b = 0$

$4 = 4$

$5 = 5$

$a = 1$

$b = 0$

$4 = 4$

$5 = 5$

$a = 1$

$b = 0$

$4 = 4$

$5 = 5$

$a = 1$

Step 1.1.3.5

Remove any equations from the system that are always true.

$b = 0$

$a = 1$

Step 1.1.3.6

List all of the solutions.

$b = 0, a = 1$

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 $a = 1$ , $b = 0$ , and $x = - 8$ .

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Step 1.1.4.1.1

Multiply $- 8$ by $1$ .

$y = - 8 + 0$

Step 1.1.4.1.2

Add $- 8$ and $0$ .

$y = - 8$

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 $y = - 8$ and $y = - 8$ .

$- 8 = - 8$

Step 1.1.4.3

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

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Step 1.1.4.3.1

Multiply $4$ by $1$ .

$y = 4 + 0$

Step 1.1.4.3.2

Add $4$ and $0$ .

$y = 4$

Step 1.1.4.4

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

$4 = 4$

Step 1.1.4.5

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

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Step 1.1.4.5.1

Multiply $5$ by $1$ .

$y = 5 + 0$

Step 1.1.4.5.2

Add $5$ and $0$ .

$y = 5$

Step 1.1.4.6

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

$5 = 5$

Step 1.1.4.7

Calculate the value of $y$ when $a = 1$ , $b = 0$ , and $x = - 3$ .

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Step 1.1.4.7.1

Multiply $- 3$ by $1$ .

$y = - 3 + 0$

Step 1.1.4.7.2

Add $- 3$ and $0$ .

$y = - 3$

Step 1.1.4.8

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 1.1.4.9

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 $y = x$ .

$y = x$

Step 2

Find $t + 1$ .

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Step 2.1

Use the function rule equation to find $t + 1$ .

$3 = t + 1$

Step 2.2

Rewrite the equation as $t + 1 = 3$ .

$t + 1 = 3$

Step 2.3

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

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Step 2.3.1

Subtract $1$ from both sides of the equation.

$t = 3 - 1$

Step 2.3.2

Subtract $1$ from $3$ .

$t = 2$

Step 3

Find $x - 7$ .

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Step 3.1

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

$x - 7 = - 6$

Step 3.2

Simplify.

$x - 7 = - 6$

Step 4

List all of the solutions.

$t = 2 x - 7 = - 6$