Examples

$[\begin{array}{c} 1 & 2 \\ - 2 & - 3 \end{array}] = [\begin{array}{c} x & 2 \\ - 2 & - 3 \end{array}]$

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

$y = a x + b$ .

$2 = a (2) + b - 2 = a (- 2) + b - 3 = a (- 3) + b$

Step 1.1.3

Calculate the values of

$a$ and

$b$ .

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Step 1.1.3.1

Solve for

$b$ in

$2 = a (2) + b$ .

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Step 1.1.3.1.1

Rewrite the equation as

$a (2) + b = 2$ .

$a (2) + b = 2$

$- 2 = a (- 2) + b$

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

Step 1.1.3.1.2

Move

$2$ to the left of

$a$ .

$2 a + b = 2$

$- 2 = a (- 2) + b$

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

Step 1.1.3.1.3

Subtract

$2 a$ from both sides of the equation.

$b = 2 - 2 a$

$- 2 = a (- 2) + b$

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

$b = 2 - 2 a$

$- 2 = a (- 2) + b$

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

Step 1.1.3.2

Replace all occurrences of

$b$ with

$2 - 2 a$ in each equation.

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Step 1.1.3.2.1

Replace all occurrences of

$b$ in

$- 2 = a (- 2) + b$ with

$2 - 2 a$ .

$- 2 = a (- 2) + 2 - 2 a$

$b = 2 - 2 a$

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

Step 1.1.3.2.2

Simplify

$- 2 = a (- 2) + 2 - 2 a$ .

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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.

$- 2 = a (- 2) + 2 - 2 a$

$b = 2 - 2 a$

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

$- 2 = a (- 2) + 2 - 2 a$

$b = 2 - 2 a$

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

Step 1.1.3.2.2.2

Simplify the right side.

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Step 1.1.3.2.2.2.1

Simplify

$a (- 2) + 2 - 2 a$ .

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Step 1.1.3.2.2.2.1.1

Move

$- 2$ to the left of

$a$ .

$- 2 = - 2 a + 2 - 2 a$

$b = 2 - 2 a$

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

Step 1.1.3.2.2.2.1.2

Subtract

$2 a$ from

$- 2 a$ .

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

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

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

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

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

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

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

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

Step 1.1.3.2.3

Replace all occurrences of

$b$ in

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

$2 - 2 a$ .

$- 3 = a (- 3) + 2 - 2 a$

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

Step 1.1.3.2.4

Simplify

$- 3 = a (- 3) + 2 - 2 a$ .

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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.

$- 3 = a (- 3) + 2 - 2 a$

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

$- 3 = a (- 3) + 2 - 2 a$

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

Step 1.1.3.2.4.2

Simplify the right side.

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Step 1.1.3.2.4.2.1

Simplify

$a (- 3) + 2 - 2 a$ .

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Step 1.1.3.2.4.2.1.1

Move

$- 3$ to the left of

$a$ .

$- 3 = - 3 a + 2 - 2 a$

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

Step 1.1.3.2.4.2.1.2

Subtract

$2 a$ from

$- 3 a$ .

$- 3 = - 5 a + 2$

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

$- 3 = - 5 a + 2$

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

$- 3 = - 5 a + 2$

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

$- 3 = - 5 a + 2$

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

$- 3 = - 5 a + 2$

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

Step 1.1.3.3

Solve for

$a$ in

$- 3 = - 5 a + 2$ .

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Step 1.1.3.3.1

Rewrite the equation as

$- 5 a + 2 = - 3$ .

$- 5 a + 2 = - 3$

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

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

$2$ from both sides of the equation.

$- 5 a = - 3 - 2$

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

Step 1.1.3.3.2.2

Subtract

$2$ from

$- 3$ .

$- 5 a = - 5$

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

$- 5 a = - 5$

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

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

$- 5$ .

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

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

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.

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

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

Step 1.1.3.3.3.2.1.2

Divide

$a$ by

$1$ .

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

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

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

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

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

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

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

$- 5$ .

$a = 1$

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

$a = 1$

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

$a = 1$

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

$a = 1$

$- 2 = - 4 a + 2$

$b = 2 - 2 a$

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

$- 2 = - 4 a + 2$ with

$1$ .

$- 2 = - 4 \cdot 1 + 2$

$a = 1$

$b = 2 - 2 a$

Step 1.1.3.4.2

Simplify the right side.

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Step 1.1.3.4.2.1

Simplify

$- 4 \cdot 1 + 2$ .

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Step 1.1.3.4.2.1.1

Multiply

$- 4$ by

$1$ .

$- 2 = - 4 + 2$

$a = 1$

$b = 2 - 2 a$

Step 1.1.3.4.2.1.2

Add

$- 4$ and

$2$ .

$- 2 = - 2$

$a = 1$

$b = 2 - 2 a$

$- 2 = - 2$

$a = 1$

$b = 2 - 2 a$

$- 2 = - 2$

$a = 1$

$b = 2 - 2 a$

Step 1.1.3.4.3

Replace all occurrences of

$a$ in

$b = 2 - 2 a$ with

$1$ .

$b = 2 - 2 \cdot 1$

$- 2 = - 2$

$a = 1$

Step 1.1.3.4.4

Simplify the right side.

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Step 1.1.3.4.4.1

Simplify

$2 - 2 \cdot 1$ .

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Step 1.1.3.4.4.1.1

Multiply

$- 2$ by

$1$ .

$b = 2 - 2$

$- 2 = - 2$

$a = 1$

Step 1.1.3.4.4.1.2

Subtract

$2$ from

$2$ .

$b = 0$

$- 2 = - 2$

$a = 1$

$b = 0$

$- 2 = - 2$

$a = 1$

$b = 0$

$- 2 = - 2$

$a = 1$

$b = 0$

$- 2 = - 2$

$a = 1$

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

$x = 2$ .

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Step 1.1.4.1.1

Multiply

$2$ by

$1$ .

$y = 2 + 0$

Step 1.1.4.1.2

Add

$2$ and

$0$ .

$y = 2$

Step 1.1.4.2

If the table has a linear function rule,

$y = y$ for the corresponding

$x$ value,

$x = 2$ . This check passes since

$y = 2$ and

$y = 2$ .

$2 = 2$

Step 1.1.4.3

Calculate the value of

$y$ when

$a = 1$ ,

$b = 0$ , and

$x = - 2$ .

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Step 1.1.4.3.1

Multiply

$- 2$ by

$1$ .

$y = - 2 + 0$

Step 1.1.4.3.2

Add

$- 2$ and

$0$ .

$y = - 2$

Step 1.1.4.4

If the table has a linear function rule,

$y = y$ for the corresponding

$x$ value,

$x = - 2$ . This check passes since

$y = - 2$ and

$y = - 2$ .

$- 2 = - 2$

Step 1.1.4.5

Calculate the value of

$y$ when

$a = 1$ ,

$b = 0$ , and

$x = - 3$ .

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Step 1.1.4.5.1

Multiply

$- 3$ by

$1$ .

$y = - 3 + 0$

Step 1.1.4.5.2

Add

$- 3$ and

$0$ .

$y = - 3$

Step 1.1.4.6

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 1.1.4.7

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

$x$ .

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Step 2.1

Use the function rule equation to find

$x$ .

$x = 1$

Step 2.2

Simplify.

$x = 1$

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