Algebra Examples

$(2, 4)$ , $(3, 9)$ , $(4, 16)$ , $(5, 25)$

Step 1

To find if the relation is linear, check to see if the values follow the linear form $y = a x + b$ .

$y = a x + b$

Step 2

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

$4 = a (2) + b 9 = a (3) + b 16 = a (4) + b 25 = a (5) + b$

Step 3

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

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Step 3.1

Solve for $b$ in $4 = a (2) + b$ .

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Step 3.1.1

Rewrite the equation as $a (2) + b = 4$ .

$a (2) + b = 4$

$9 = a (3) + b$

$16 = a (4) + b$

$25 = a (5) + b$

Step 3.1.2

Move $2$ to the left of $a$ .

$2 a + b = 4$

$9 = a (3) + b$

$16 = a (4) + b$

$25 = a (5) + b$

Step 3.1.3

Subtract $2 a$ from both sides of the equation.

$b = 4 - 2 a$

$9 = a (3) + b$

$16 = a (4) + b$

$25 = a (5) + b$

$b = 4 - 2 a$

$9 = a (3) + b$

$16 = a (4) + b$

$25 = a (5) + b$

Step 3.2

Replace all occurrences of $b$ with $4 - 2 a$ in each equation.

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Step 3.2.1

Replace all occurrences of $b$ in $9 = a (3) + b$ with $4 - 2 a$ .

$9 = a (3) + 4 - 2 a$

$b = 4 - 2 a$

$16 = a (4) + b$

$25 = a (5) + b$

Step 3.2.2

Simplify $9 = a (3) + 4 - 2 a$ .

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Step 3.2.2.1

Simplify the left side.

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Step 3.2.2.1.1

Remove parentheses.

$9 = a (3) + 4 - 2 a$

$b = 4 - 2 a$

$16 = a (4) + b$

$25 = a (5) + b$

$9 = a (3) + 4 - 2 a$

$b = 4 - 2 a$

$16 = a (4) + b$

$25 = a (5) + b$

Step 3.2.2.2

Simplify the right side.

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Step 3.2.2.2.1

Simplify $a (3) + 4 - 2 a$ .

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Step 3.2.2.2.1.1

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

$9 = 3 a + 4 - 2 a$

$b = 4 - 2 a$

$16 = a (4) + b$

$25 = a (5) + b$

Step 3.2.2.2.1.2

Subtract $2 a$ from $3 a$ .

$9 = a + 4$

$b = 4 - 2 a$

$16 = a (4) + b$

$25 = a (5) + b$

$9 = a + 4$

$b = 4 - 2 a$

$16 = a (4) + b$

$25 = a (5) + b$

$9 = a + 4$

$b = 4 - 2 a$

$16 = a (4) + b$

$25 = a (5) + b$

$9 = a + 4$

$b = 4 - 2 a$

$16 = a (4) + b$

$25 = a (5) + b$

Step 3.2.3

Replace all occurrences of $b$ in $16 = a (4) + b$ with $4 - 2 a$ .

$16 = a (4) + 4 - 2 a$

$9 = a + 4$

$b = 4 - 2 a$

$25 = a (5) + b$

Step 3.2.4

Simplify $16 = a (4) + 4 - 2 a$ .

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Step 3.2.4.1

Simplify the left side.

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Step 3.2.4.1.1

Remove parentheses.

$16 = a (4) + 4 - 2 a$

$9 = a + 4$

$b = 4 - 2 a$

$25 = a (5) + b$

$16 = a (4) + 4 - 2 a$

$9 = a + 4$

$b = 4 - 2 a$

$25 = a (5) + b$

Step 3.2.4.2

Simplify the right side.

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Step 3.2.4.2.1

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

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Step 3.2.4.2.1.1

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

$16 = 4 a + 4 - 2 a$

$9 = a + 4$

$b = 4 - 2 a$

$25 = a (5) + b$

Step 3.2.4.2.1.2

Subtract $2 a$ from $4 a$ .

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

$25 = a (5) + b$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

$25 = a (5) + b$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

$25 = a (5) + b$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

$25 = a (5) + b$

Step 3.2.5

Replace all occurrences of $b$ in $25 = a (5) + b$ with $4 - 2 a$ .

$25 = a (5) + 4 - 2 a$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

Step 3.2.6

Simplify $25 = a (5) + 4 - 2 a$ .

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Step 3.2.6.1

Simplify the left side.

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Step 3.2.6.1.1

Remove parentheses.

$25 = a (5) + 4 - 2 a$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

$25 = a (5) + 4 - 2 a$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

Step 3.2.6.2

Simplify the right side.

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Step 3.2.6.2.1

Simplify $a (5) + 4 - 2 a$ .

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Step 3.2.6.2.1.1

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

$25 = 5 a + 4 - 2 a$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

Step 3.2.6.2.1.2

Subtract $2 a$ from $5 a$ .

$25 = 3 a + 4$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

$25 = 3 a + 4$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

$25 = 3 a + 4$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

$25 = 3 a + 4$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

$25 = 3 a + 4$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

Step 3.3

Solve for $a$ in $25 = 3 a + 4$ .

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Step 3.3.1

Rewrite the equation as $3 a + 4 = 25$ .

$3 a + 4 = 25$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $4$ from both sides of the equation.

$3 a = 25 - 4$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

Step 3.3.2.2

Subtract $4$ from $25$ .

$3 a = 21$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

$3 a = 21$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

Step 3.3.3

Divide each term in $3 a = 21$ by $3$ and simplify.

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Step 3.3.3.1

Divide each term in $3 a = 21$ by $3$ .

$\frac{3 a}{3} = \frac{21}{3}$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $3$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{3 a}{3} = \frac{21}{3}$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

Step 3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{21}{3}$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

$a = \frac{21}{3}$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

$a = \frac{21}{3}$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $21$ by $3$ .

$a = 7$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

$a = 7$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

$a = 7$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

$a = 7$

$16 = 2 a + 4$

$9 = a + 4$

$b = 4 - 2 a$

Step 3.4

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

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Step 3.4.1

Replace all occurrences of $a$ in $16 = 2 a + 4$ with $7$ .

$16 = 2 (7) + 4$

$a = 7$

$9 = a + 4$

$b = 4 - 2 a$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $2 (7) + 4$ .

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Step 3.4.2.1.1

Multiply $2$ by $7$ .

$16 = 14 + 4$

$a = 7$

$9 = a + 4$

$b = 4 - 2 a$

Step 3.4.2.1.2

Add $14$ and $4$ .

$16 = 18$

$a = 7$

$9 = a + 4$

$b = 4 - 2 a$

$16 = 18$

$a = 7$

$9 = a + 4$

$b = 4 - 2 a$

$16 = 18$

$a = 7$

$9 = a + 4$

$b = 4 - 2 a$

Step 3.4.3

Replace all occurrences of $a$ in $9 = a + 4$ with $7$ .

$9 = (7) + 4$

$16 = 18$

$a = 7$

$b = 4 - 2 a$

Step 3.4.4

Simplify the right side.

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Step 3.4.4.1

Add $7$ and $4$ .

$9 = 11$

$16 = 18$

$a = 7$

$b = 4 - 2 a$

$9 = 11$

$16 = 18$

$a = 7$

$b = 4 - 2 a$

Step 3.4.5

Replace all occurrences of $a$ in $b = 4 - 2 a$ with $7$ .

$b = 4 - 2 \cdot 7$

$9 = 11$

$16 = 18$

$a = 7$

Step 3.4.6

Simplify the right side.

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Step 3.4.6.1

Simplify $4 - 2 \cdot 7$ .

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Step 3.4.6.1.1

Multiply $- 2$ by $7$ .

$b = 4 - 14$

$9 = 11$

$16 = 18$

$a = 7$

Step 3.4.6.1.2

Subtract $14$ from $4$ .

$b = - 10$

$9 = 11$

$16 = 18$

$a = 7$

$b = - 10$

$9 = 11$

$16 = 18$

$a = 7$

$b = - 10$

$9 = 11$

$16 = 18$

$a = 7$

$b = - 10$

$9 = 11$

$16 = 18$

$a = 7$

Step 3.5

Since $9 = 11$ is not true, there is no solution.

No solution

Step 4

Since $y \neq y$ for the corresponding $x$ values, the relation is not linear.

The relation is not linear