Algebra Examples

$\begin{array}{c} x & y \\ 0 & 7 \\ 1 & 9 \end{array}$

Step 1

Check if the function rule is linear.

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Step 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 1.2

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

$7 = a (0) + b 9 = a (1) + b$

Step 1.3

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

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Step 1.3.1

Rewrite the equation as $b = 7$ .

$b = 7$

$9 = a + b$

Step 1.3.2

Replace all occurrences of $b$ with $7$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $b$ in $9 = a + b$ with $7$ .

$9 = a + 7$

$b = 7$

Step 1.3.2.2

Simplify the left side.

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Step 1.3.2.2.1

Remove parentheses.

$9 = a + 7$

$b = 7$

$9 = a + 7$

$b = 7$

$9 = a + 7$

$b = 7$

Step 1.3.3

Solve for $a$ in $9 = a + 7$ .

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Step 1.3.3.1

Rewrite the equation as $a + 7 = 9$ .

$a + 7 = 9$

$b = 7$

Step 1.3.3.2

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

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Step 1.3.3.2.1

Subtract $7$ from both sides of the equation.

$a = 9 - 7$

$b = 7$

Step 1.3.3.2.2

Subtract $7$ from $9$ .

$a = 2$

$b = 7$

$a = 2$

$b = 7$

$a = 2$

$b = 7$

Step 1.3.4

Solve the system of equations.

$a = 2 b = 7$

Step 1.3.5

List all of the solutions.

$a = 2, b = 7$

Step 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 1.4.1

Calculate the value of $y$ when $a = 2$ , $b = 7$ , and $x = 0$ .

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Step 1.4.1.1

Multiply $2$ by $0$ .

$y = 0 + 7$

Step 1.4.1.2

Add $0$ and $7$ .

$y = 7$

Step 1.4.2

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 0$ . This check passes since $y = 7$ and $y = 7$ .

$7 = 7$

Step 1.4.3

Calculate the value of $y$ when $a = 2$ , $b = 7$ , and $x = 1$ .

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Step 1.4.3.1

Multiply $2$ by $1$ .

$y = 2 + 7$

Step 1.4.3.2

Add $2$ and $7$ .

$y = 9$

Step 1.4.4

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 1$ . This check passes since $y = 9$ and $y = 9$ .

$9 = 9$

Step 1.4.5

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

The function is linear

Step 2

Since all $y = y$ , the function is linear and follows the form $y = 2 x + 7$ .

$y = 2 x + 7$