Algebra Examples

\begin{matrix} x & q (x) 2 & 4 4 & 6 6 & 8 7 & 9 \end{matrix}

$\begin{array}{c} x & q (x) \\ 2 & 4 \\ 4 & 6 \\ 6 & 8 \\ 7 & 9 \end{array}$

Step 1

Check if the function rule is linear.

Tap for more steps...

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

$y = a x + b$ .

y = a x + b

$y = a x + b$

Step 1.2

Build a set of equations from the table such that

q (x) = a x + b

$q (x) = a x + b$ .

$4 = a (2) + b 6 = a (4) + b 8 = a (6) + b 9 = a (7) + b$

Step 1.3

Calculate the values of

$a$ and

$b$ .

Tap for more steps...

Step 1.3.1

Solve for

$b$ in

$4 = a (2) + b$ .

Tap for more steps...

Step 1.3.1.1

Rewrite the equation as

$a (2) + b = 4$ .

$a (2) + b = 4$

$6 = a (4) + b$

$8 = a (6) + b$

$9 = a (7) + b$

Step 1.3.1.2

Move

$2$ to the left of

$a$ .

$2 a + b = 4$

$6 = a (4) + b$

$8 = a (6) + b$

$9 = a (7) + b$

Step 1.3.1.3

Subtract

$2 a$ from both sides of the equation.

$b = 4 - 2 a$

$6 = a (4) + b$

$8 = a (6) + b$

$9 = a (7) + b$

$b = 4 - 2 a$

$6 = a (4) + b$

$8 = a (6) + b$

$9 = a (7) + b$

Step 1.3.2

Replace all occurrences of

$b$ with

$4 - 2 a$ in each equation.

Tap for more steps...

Step 1.3.2.1

Replace all occurrences of

$b$ in

$6 = a (4) + b$ with

$4 - 2 a$ .

$6 = a (4) + 4 - 2 a$

$b = 4 - 2 a$

$8 = a (6) + b$

$9 = a (7) + b$

Step 1.3.2.2

Simplify

$6 = a (4) + 4 - 2 a$ .

Tap for more steps...

Step 1.3.2.2.1

Simplify the left side.

Tap for more steps...

Step 1.3.2.2.1.1

Remove parentheses.

$6 = a (4) + 4 - 2 a$

$b = 4 - 2 a$

$8 = a (6) + b$

$9 = a (7) + b$

$6 = a (4) + 4 - 2 a$

$b = 4 - 2 a$

$8 = a (6) + b$

$9 = a (7) + b$

Step 1.3.2.2.2

Simplify the right side.

Tap for more steps...

Step 1.3.2.2.2.1

Simplify

$a (4) + 4 - 2 a$ .

Tap for more steps...

Step 1.3.2.2.2.1.1

Move

$4$ to the left of

$a$ .

$6 = 4 a + 4 - 2 a$

$b = 4 - 2 a$

$8 = a (6) + b$

$9 = a (7) + b$

Step 1.3.2.2.2.1.2

Subtract

$2 a$ from

$4 a$ .

$6 = 2 a + 4$

$b = 4 - 2 a$

$8 = a (6) + b$

$9 = a (7) + b$

$6 = 2 a + 4$

$b = 4 - 2 a$

$8 = a (6) + b$

$9 = a (7) + b$

$6 = 2 a + 4$

$b = 4 - 2 a$

$8 = a (6) + b$

$9 = a (7) + b$

$6 = 2 a + 4$

$b = 4 - 2 a$

$8 = a (6) + b$

$9 = a (7) + b$

Step 1.3.2.3

Replace all occurrences of

$b$ in

$8 = a (6) + b$ with

$4 - 2 a$ .

$8 = a (6) + 4 - 2 a$

$6 = 2 a + 4$

$b = 4 - 2 a$

$9 = a (7) + b$

Step 1.3.2.4

Simplify

$8 = a (6) + 4 - 2 a$ .

Tap for more steps...

Step 1.3.2.4.1

Simplify the left side.

Tap for more steps...

Step 1.3.2.4.1.1

Remove parentheses.

$8 = a (6) + 4 - 2 a$

$6 = 2 a + 4$

$b = 4 - 2 a$

$9 = a (7) + b$

$8 = a (6) + 4 - 2 a$

$6 = 2 a + 4$

$b = 4 - 2 a$

$9 = a (7) + b$

Step 1.3.2.4.2

Simplify the right side.

Tap for more steps...

Step 1.3.2.4.2.1

Simplify

$a (6) + 4 - 2 a$ .

Tap for more steps...

Step 1.3.2.4.2.1.1

Move

$6$ to the left of

$a$ .

$8 = 6 a + 4 - 2 a$

$6 = 2 a + 4$

$b = 4 - 2 a$

$9 = a (7) + b$

Step 1.3.2.4.2.1.2

Subtract

$2 a$ from

$6 a$ .

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

$9 = a (7) + b$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

$9 = a (7) + b$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

$9 = a (7) + b$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

$9 = a (7) + b$

Step 1.3.2.5

Replace all occurrences of

$b$ in

$9 = a (7) + b$ with

$4 - 2 a$ .

$9 = a (7) + 4 - 2 a$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

Step 1.3.2.6

Simplify

$9 = a (7) + 4 - 2 a$ .

Tap for more steps...

Step 1.3.2.6.1

Simplify the left side.

Tap for more steps...

Step 1.3.2.6.1.1

Remove parentheses.

$9 = a (7) + 4 - 2 a$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

$9 = a (7) + 4 - 2 a$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

Step 1.3.2.6.2

Simplify the right side.

Tap for more steps...

Step 1.3.2.6.2.1

Simplify

$a (7) + 4 - 2 a$ .

Tap for more steps...

Step 1.3.2.6.2.1.1

Move

$7$ to the left of

$a$ .

$9 = 7 a + 4 - 2 a$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

Step 1.3.2.6.2.1.2

Subtract

$2 a$ from

$7 a$ .

$9 = 5 a + 4$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

$9 = 5 a + 4$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

$9 = 5 a + 4$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

$9 = 5 a + 4$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

$9 = 5 a + 4$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

Step 1.3.3

Solve for

$a$ in

$9 = 5 a + 4$ .

Tap for more steps...

Step 1.3.3.1

Rewrite the equation as

$5 a + 4 = 9$ .

$5 a + 4 = 9$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

Step 1.3.3.2

Move all terms not containing

$a$ to the right side of the equation.

Tap for more steps...

Step 1.3.3.2.1

Subtract

$4$ from both sides of the equation.

$5 a = 9 - 4$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

Step 1.3.3.2.2

Subtract

$4$ from

$9$ .

$5 a = 5$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

$5 a = 5$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

Step 1.3.3.3

Divide each term in

$5 a = 5$ by

$5$ and simplify.

Tap for more steps...

Step 1.3.3.3.1

Divide each term in

$5 a = 5$ by

$5$ .

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

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

Step 1.3.3.3.2

Simplify the left side.

Tap for more steps...

Step 1.3.3.3.2.1

Cancel the common factor of

$5$ .

Tap for more steps...

Step 1.3.3.3.2.1.1

Cancel the common factor.

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

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

Step 1.3.3.3.2.1.2

Divide

$a$ by

$1$ .

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

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

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

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

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

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

Step 1.3.3.3.3

Simplify the right side.

Tap for more steps...

Step 1.3.3.3.3.1

Divide

$5$ by

$5$ .

$a = 1$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

$a = 1$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

$a = 1$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

$a = 1$

$8 = 4 a + 4$

$6 = 2 a + 4$

$b = 4 - 2 a$

Step 1.3.4

Replace all occurrences of

$a$ with

$1$ in each equation.

Tap for more steps...

Step 1.3.4.1

Replace all occurrences of

$a$ in

$8 = 4 a + 4$ with

$1$ .

$8 = 4 (1) + 4$

$a = 1$

$6 = 2 a + 4$

$b = 4 - 2 a$

Step 1.3.4.2

Simplify the right side.

Tap for more steps...

Step 1.3.4.2.1

Simplify

$4 (1) + 4$ .

Tap for more steps...

Step 1.3.4.2.1.1

Multiply

$4$ by

$1$ .

$8 = 4 + 4$

$a = 1$

$6 = 2 a + 4$

$b = 4 - 2 a$

Step 1.3.4.2.1.2

Add

$4$ and

$4$ .

$8 = 8$

$a = 1$

$6 = 2 a + 4$

$b = 4 - 2 a$

$8 = 8$

$a = 1$

$6 = 2 a + 4$

$b = 4 - 2 a$

$8 = 8$

$a = 1$

$6 = 2 a + 4$

$b = 4 - 2 a$

Step 1.3.4.3

Replace all occurrences of

$a$ in

$6 = 2 a + 4$ with

$1$ .

$6 = 2 (1) + 4$

$8 = 8$

$a = 1$

$b = 4 - 2 a$

Step 1.3.4.4

Simplify the right side.

Tap for more steps...

Step 1.3.4.4.1

Simplify

$2 (1) + 4$ .

Tap for more steps...

Step 1.3.4.4.1.1

Multiply

$2$ by

$1$ .

$6 = 2 + 4$

$8 = 8$

$a = 1$

$b = 4 - 2 a$

Step 1.3.4.4.1.2

Add

$2$ and

$4$ .

$6 = 6$

$8 = 8$

$a = 1$

$b = 4 - 2 a$

$6 = 6$

$8 = 8$

$a = 1$

$b = 4 - 2 a$

$6 = 6$

$8 = 8$

$a = 1$

$b = 4 - 2 a$

Step 1.3.4.5

Replace all occurrences of

$a$ in

$b = 4 - 2 a$ with

$1$ .

$b = 4 - 2 \cdot 1$

$6 = 6$

$8 = 8$

$a = 1$

Step 1.3.4.6

Simplify the right side.

Tap for more steps...

Step 1.3.4.6.1

Simplify

$4 - 2 \cdot 1$ .

Tap for more steps...

Step 1.3.4.6.1.1

Multiply

$- 2$ by

$1$ .

$b = 4 - 2$

$6 = 6$

$8 = 8$

$a = 1$

Step 1.3.4.6.1.2

Subtract

$2$ from

$4$ .

$b = 2$

$6 = 6$

$8 = 8$

$a = 1$

$b = 2$

$6 = 6$

$8 = 8$

$a = 1$

$b = 2$

$6 = 6$

$8 = 8$

$a = 1$

$b = 2$

$6 = 6$

$8 = 8$

$a = 1$

Step 1.3.5

Remove any equations from the system that are always true.

$b = 2$

$a = 1$

Step 1.3.6

List all of the solutions.

$b = 2, a = 1$

Step 1.4

Calculate the value of

$y$ using each

$x$ value in the relation and compare this value to the given

$q (x)$ value in the relation.

Tap for more steps...

Step 1.4.1

Calculate the value of

$y$ when

$a = 1$ ,

$b = 2$ , and

$x = 2$ .

Tap for more steps...

Step 1.4.1.1

Multiply

$2$ by

$1$ .

$y = 2 + 2$

Step 1.4.1.2

Add

$2$ and

$2$ .

$y = 4$

Step 1.4.2

If the table has a linear function rule,

$y = q (x)$ for the corresponding

$x$ value,

$x = 2$ . This check passes since

$y = 4$ and

$q (x) = 4$ .

$4 = 4$

Step 1.4.3

Calculate the value of

$y$ when

$a = 1$ ,

$b = 2$ , and

$x = 4$ .

Tap for more steps...

Step 1.4.3.1

Multiply

$4$ by

$1$ .

$y = 4 + 2$

Step 1.4.3.2

Add

$4$ and

$2$ .

$y = 6$

Step 1.4.4

If the table has a linear function rule,

$y = q (x)$ for the corresponding

$x$ value,

$x = 4$ . This check passes since

$y = 6$ and

$q (x) = 6$ .

$6 = 6$

Step 1.4.5

Calculate the value of

$y$ when

$a = 1$ ,

$b = 2$ , and

$x = 6$ .

Tap for more steps...

Step 1.4.5.1

Multiply

$6$ by

$1$ .

$y = 6 + 2$

Step 1.4.5.2

Add

$6$ and

$2$ .

$y = 8$

Step 1.4.6

If the table has a linear function rule,

$y = q (x)$ for the corresponding

$x$ value,

$x = 6$ . This check passes since

$y = 8$ and

$q (x) = 8$ .

$8 = 8$

Step 1.4.7

Calculate the value of

$y$ when

$a = 1$ ,

$b = 2$ , and

$x = 7$ .

Tap for more steps...

Step 1.4.7.1

Multiply

$7$ by

$1$ .

$y = 7 + 2$

Step 1.4.7.2

Add

$7$ and

$2$ .

$y = 9$

Step 1.4.8

If the table has a linear function rule,

$y = q (x)$ for the corresponding

$x$ value,

$x = 7$ . This check passes since

$y = 9$ and

$q (x) = 9$ .

$9 = 9$

Step 1.4.9

Since

$y = q (x)$ for the corresponding

$x$ values, the function is linear.

The function is linear

Step 2

Since all

$y = q (x)$ , the function is linear and follows the form

$y = x + 2$ .

$y = x + 2$

Enter YOUR Problem