Precalculus Examples

$[\begin{array}{c} x + 1 & 2 & 3 \\ 4 & y - 1 & 5 \\ u & - 1 & z + 2 \end{array}] = [\begin{array}{c} 2 x - 1 & t + 1 & 3 \\ v + 1 & - 3 & 5 \\ - 4 & w - 1 & 2 z - 1 \end{array}]$

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 $y = a x + b$ .

$3 = a (3) + b 5 = a (5) + b$

Step 1.1.3

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

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Step 1.1.3.1

Solve for $b$ in $3 = a (3) + b$ .

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Step 1.1.3.1.1

Rewrite the equation as $a (3) + b = 3$ .

$a (3) + b = 3$

$5 = a (5) + b$

Step 1.1.3.1.2

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

$3 a + b = 3$

$5 = a (5) + b$

Step 1.1.3.1.3

Subtract $3 a$ from both sides of the equation.

$b = 3 - 3 a$

$5 = a (5) + b$

$b = 3 - 3 a$

$5 = a (5) + b$

Step 1.1.3.2

Replace all occurrences of $b$ with $3 - 3 a$ in each equation.

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Step 1.1.3.2.1

Replace all occurrences of $b$ in $5 = a (5) + b$ with $3 - 3 a$ .

$5 = a (5) + 3 - 3 a$

$b = 3 - 3 a$

Step 1.1.3.2.2

Simplify $5 = a (5) + 3 - 3 a$ .

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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.

$5 = a (5) + 3 - 3 a$

$b = 3 - 3 a$

$5 = a (5) + 3 - 3 a$

$b = 3 - 3 a$

Step 1.1.3.2.2.2

Simplify the right side.

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Step 1.1.3.2.2.2.1

Simplify $a (5) + 3 - 3 a$ .

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Step 1.1.3.2.2.2.1.1

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

$5 = 5 a + 3 - 3 a$

$b = 3 - 3 a$

Step 1.1.3.2.2.2.1.2

Subtract $3 a$ from $5 a$ .

$5 = 2 a + 3$

$b = 3 - 3 a$

$5 = 2 a + 3$

$b = 3 - 3 a$

$5 = 2 a + 3$

$b = 3 - 3 a$

$5 = 2 a + 3$

$b = 3 - 3 a$

$5 = 2 a + 3$

$b = 3 - 3 a$

Step 1.1.3.3

Solve for $a$ in $5 = 2 a + 3$ .

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Step 1.1.3.3.1

Rewrite the equation as $2 a + 3 = 5$ .

$2 a + 3 = 5$

$b = 3 - 3 a$

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 $3$ from both sides of the equation.

$2 a = 5 - 3$

$b = 3 - 3 a$

Step 1.1.3.3.2.2

Subtract $3$ from $5$ .

$2 a = 2$

$b = 3 - 3 a$

$2 a = 2$

$b = 3 - 3 a$

Step 1.1.3.3.3

Divide each term in $2 a = 2$ by $2$ and simplify.

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Step 1.1.3.3.3.1

Divide each term in $2 a = 2$ by $2$ .

$\frac{2 a}{2} = \frac{2}{2}$

$b = 3 - 3 a$

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 $2$ .

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Step 1.1.3.3.3.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{2}{2}$

$b = 3 - 3 a$

Step 1.1.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{2}{2}$

$b = 3 - 3 a$

$a = \frac{2}{2}$

$b = 3 - 3 a$

$a = \frac{2}{2}$

$b = 3 - 3 a$

Step 1.1.3.3.3.3

Simplify the right side.

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Step 1.1.3.3.3.3.1

Divide $2$ by $2$ .

$a = 1$

$b = 3 - 3 a$

$a = 1$

$b = 3 - 3 a$

$a = 1$

$b = 3 - 3 a$

$a = 1$

$b = 3 - 3 a$

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 $b = 3 - 3 a$ with $1$ .

$b = 3 - 3 \cdot 1$

$a = 1$

Step 1.1.3.4.2

Simplify the right side.

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Step 1.1.3.4.2.1

Simplify $3 - 3 \cdot 1$ .

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Step 1.1.3.4.2.1.1

Multiply $- 3$ by $1$ .

$b = 3 - 3$

$a = 1$

Step 1.1.3.4.2.1.2

Subtract $3$ from $3$ .

$b = 0$

$a = 1$

$b = 0$

$a = 1$

$b = 0$

$a = 1$

$b = 0$

$a = 1$

Step 1.1.3.5

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 $x = 3$ .

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Step 1.1.4.1.1

Multiply $3$ by $1$ .

$y = 3 + 0$

Step 1.1.4.1.2

Add $3$ and $0$ .

$y = 3$

Step 1.1.4.2

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 1.1.4.3

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

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Step 1.1.4.3.1

Multiply $5$ by $1$ .

$y = 5 + 0$

Step 1.1.4.3.2

Add $5$ and $0$ .

$y = 5$

Step 1.1.4.4

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 1.1.4.5

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 $y = x$ .

$y = x$

Step 2

Find $x + 1$ .

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Step 2.1

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

$2 x + 1 - 1 = x + 1$

Step 2.2

Combine the opposite terms in $2 x + 1 - 1$ .

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Step 2.2.1

Subtract $1$ from $1$ .

$2 x + 0 = x + 1$

Step 2.2.2

Add $2 x$ and $0$ .

$2 x = x + 1$

Step 2.3

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

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Step 2.3.1

Subtract $x$ from both sides of the equation.

$2 x - x = 1$

Step 2.3.2

Subtract $x$ from $2 x$ .

$x = 1$

Step 3

Find $y - 1$ .

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Step 3.1

Use the function rule equation to find $y - 1$ .

$- 3 = y - 1$

Step 3.2

Rewrite the equation as $y - 1 = - 3$ .

$y - 1 = - 3$

Step 3.3

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

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Step 3.3.1

Add $1$ to both sides of the equation.

$y = - 3 + 1$

Step 3.3.2

Add $- 3$ and $1$ .

$y = - 2$

Step 4

Find $u$ .

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Step 4.1

Use the function rule equation to find $u$ .

$- 4 = u$

Step 4.2

Rewrite the equation as $u = - 4$ .

$u = - 4$

Step 5

Find $z + 2$ .

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Step 5.1

Use the function rule equation to find $z + 2$ .

$2 z - 1 = z + 2$

Step 5.2

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

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Step 5.2.1

Subtract $z$ from both sides of the equation.

$2 z - 1 - z = 2$

Step 5.2.2

Subtract $z$ from $2 z$ .

$z - 1 = 2$

Step 5.3

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

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Step 5.3.1

Add $1$ to both sides of the equation.

$z = 2 + 1$

Step 5.3.2

Add $2$ and $1$ .

$z = 3$

Step 6

Find $t + 1$ .

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Step 6.1

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

$t + 1 = 2$

Step 6.2

Simplify.

$t + 1 = 2$

Step 7

Find $v + 1$ .

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Step 7.1

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

$v + 1 = 4$

Step 7.2

Simplify.

$v + 1 = 4$

Step 8

Find $w - 1$ .

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Step 8.1

Use the function rule equation to find $w - 1$ .

$w - 1 = - 1$

Step 8.2

Simplify.

$w - 1 = - 1$

Step 9

List all of the solutions.

$x = 1 y = - 2 u = - 4 z = 3 t + 1 = 2 v + 1 = 4 w - 1 = - 1$