Algebra Examples

$(8, 9)$ , $(16, y)$

Step 1

Find the slope of the line between $(8, 9)$ and $(16, y)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

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Step 1.1

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

Step 1.2

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 1.3

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{y - (9)}{16 - (8)}$

Step 1.4

Simplify.

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Step 1.4.1

Multiply $- 1$ by $9$ .

$m = \frac{y - 9}{16 - (8)}$

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Multiply $- 1$ by $8$ .

$m = \frac{y - 9}{16 - 8}$

Step 1.4.2.2

Subtract $8$ from $16$ .

$m = \frac{y - 9}{8}$

Step 2

Use the slope $\frac{y - 9}{8}$ and a given point $(8, 9)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (9) = \frac{y - 9}{8} \cdot (x - (8))$

Step 3

Simplify the equation and keep it in point-slope form.

$y - 9 = \frac{y - 9}{8} \cdot (x - 8)$

Step 4

Solve for $y$ .

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Step 4.1

Simplify $\frac{y - 9}{8} \cdot (x - 8)$ .

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Step 4.1.1

Rewrite.

$y - 9 = 0 + 0 + \frac{y - 9}{8} \cdot (x - 8)$

Step 4.1.2

Simplify terms.

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Step 4.1.2.1

Apply the distributive property.

$y - 9 = \frac{y - 9}{8} x + \frac{y - 9}{8} \cdot - 8$

Step 4.1.2.2

Combine $\frac{y - 9}{8}$ and $x$ .

$y - 9 = \frac{(y - 9) x}{8} + \frac{y - 9}{8} \cdot - 8$

Step 4.1.2.3

Cancel the common factor of $8$ .

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Step 4.1.2.3.1

Factor $8$ out of $- 8$ .

$y - 9 = \frac{(y - 9) x}{8} + \frac{y - 9}{8} \cdot (8 (- 1))$

Step 4.1.2.3.2

Cancel the common factor.

$y - 9 = \frac{(y - 9) x}{8} + \frac{y - 9}{8} \cdot (8 \cdot - 1)$

Step 4.1.2.3.3

Rewrite the expression.

$y - 9 = \frac{(y - 9) x}{8} + (y - 9) \cdot - 1$

Step 4.1.3

Simplify each term.

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Step 4.1.3.1

Apply the distributive property.

$y - 9 = \frac{(y - 9) x}{8} + y \cdot - 1 - 9 \cdot - 1$

Step 4.1.3.2

Move $- 1$ to the left of $y$ .

$y - 9 = \frac{(y - 9) x}{8} - 1 \cdot y - 9 \cdot - 1$

Step 4.1.3.3

Multiply $- 9$ by $- 1$ .

$y - 9 = \frac{(y - 9) x}{8} - 1 \cdot y + 9$

Step 4.1.3.4

Rewrite $- 1 y$ as $- y$ .

$y - 9 = \frac{(y - 9) x}{8} - y + 9$

Step 4.1.4

To write $- y$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y - 9 = \frac{(y - 9) x}{8} - y \cdot \frac{8}{8} + 9$

Step 4.1.5

Simplify terms.

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Step 4.1.5.1

Combine $- y$ and $\frac{8}{8}$ .

$y - 9 = \frac{(y - 9) x}{8} + \frac{- y \cdot 8}{8} + 9$

Step 4.1.5.2

Combine the numerators over the common denominator.

$y - 9 = \frac{(y - 9) x - y \cdot 8}{8} + 9$

Step 4.1.6

Simplify the numerator.

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Step 4.1.6.1

Apply the distributive property.

$y - 9 = \frac{y x - 9 x - y \cdot 8}{8} + 9$

Step 4.1.6.2

Multiply $8$ by $- 1$ .

$y - 9 = \frac{y x - 9 x - 8 y}{8} + 9$

Step 4.1.7

To write $9$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y - 9 = \frac{y x - 9 x - 8 y}{8} + 9 \cdot \frac{8}{8}$

Step 4.1.8

Simplify terms.

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Step 4.1.8.1

Combine $9$ and $\frac{8}{8}$ .

$y - 9 = \frac{y x - 9 x - 8 y}{8} + \frac{9 \cdot 8}{8}$

Step 4.1.8.2

Combine the numerators over the common denominator.

$y - 9 = \frac{y x - 9 x - 8 y + 9 \cdot 8}{8}$

Step 4.1.9

Simplify the numerator.

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Step 4.1.9.1

Multiply $9$ by $8$ .

$y - 9 = \frac{y x - 9 x - 8 y + 72}{8}$

Step 4.1.9.2

Rewrite $y x - 9 x - 8 y + 72$ in a factored form.

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Step 4.1.9.2.1

Factor out the greatest common factor from each group.

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Step 4.1.9.2.1.1

Group the first two terms and the last two terms.

$y - 9 = \frac{(y x - 9 x) - 8 y + 72}{8}$

Step 4.1.9.2.1.2

Factor out the greatest common factor (GCF) from each group.

$y - 9 = \frac{x (y - 9) - 8 (y - 9)}{8}$

Step 4.1.9.2.2

Factor the polynomial by factoring out the greatest common factor, $y - 9$ .

$y - 9 = \frac{(y - 9) (x - 8)}{8}$

Step 4.2

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

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Step 4.2.1

Subtract $\frac{(y - 9) (x - 8)}{8}$ from both sides of the equation.

$y - 9 - \frac{(y - 9) (x - 8)}{8} = 0$

Step 4.2.2

To write $y$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y \cdot \frac{8}{8} - \frac{(y - 9) (x - 8)}{8} - 9 = 0$

Step 4.2.3

Combine $y$ and $\frac{8}{8}$ .

$\frac{y \cdot 8}{8} - \frac{(y - 9) (x - 8)}{8} - 9 = 0$

Step 4.2.4

Combine the numerators over the common denominator.

$\frac{y \cdot 8 - (y - 9) (x - 8)}{8} - 9 = 0$

Step 4.2.5

Simplify each term.

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Step 4.2.5.1

Simplify the numerator.

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Step 4.2.5.1.1

Move $8$ to the left of $y$ .

$\frac{8 \cdot y - (y - 9) (x - 8)}{8} - 9 = 0$

Step 4.2.5.1.2

Apply the distributive property.

$\frac{8 y + (- y - - 9) (x - 8)}{8} - 9 = 0$

Step 4.2.5.1.3

Multiply $- 1$ by $- 9$ .

$\frac{8 y + (- y + 9) (x - 8)}{8} - 9 = 0$

Step 4.2.5.1.4

Expand $(- y + 9) (x - 8)$ using the FOIL Method.

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Step 4.2.5.1.4.1

Apply the distributive property.

$\frac{8 y - y (x - 8) + 9 (x - 8)}{8} - 9 = 0$

Step 4.2.5.1.4.2

Apply the distributive property.

$\frac{8 y - y x - y \cdot - 8 + 9 (x - 8)}{8} - 9 = 0$

Step 4.2.5.1.4.3

Apply the distributive property.

$\frac{8 y - y x - y \cdot - 8 + 9 x + 9 \cdot - 8}{8} - 9 = 0$

Step 4.2.5.1.5

Simplify each term.

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Step 4.2.5.1.5.1

Multiply $- 8$ by $- 1$ .

$\frac{8 y - y x + 8 y + 9 x + 9 \cdot - 8}{8} - 9 = 0$

Step 4.2.5.1.5.2

Multiply $9$ by $- 8$ .

$\frac{8 y - y x + 8 y + 9 x - 72}{8} - 9 = 0$

Step 4.2.5.1.6

Add $8 y$ and $8 y$ .

$\frac{- y x + 16 y + 9 x - 72}{8} - 9 = 0$

Step 4.2.5.1.7

Rewrite $- y x + 16 y + 9 x - 72$ in a factored form.

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Step 4.2.5.1.7.1

Regroup terms.

$\frac{- 72 - y x + 16 y + 9 x}{8} - 9 = 0$

Step 4.2.5.1.7.2

Factor $- 1$ out of $- y x + 16 y + 9 x$ .

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Step 4.2.5.1.7.2.1

Reorder the expression.

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Step 4.2.5.1.7.2.1.1

Move $y$ .

$\frac{- 72 - 1 x y + 16 y + 9 x}{8} - 9 = 0$

Step 4.2.5.1.7.2.1.2

Move $16 y$ .

$\frac{- 72 - 1 x y + 9 x + 16 y}{8} - 9 = 0$

Step 4.2.5.1.7.2.2

Factor $- 1$ out of $- 1 x y$ .

$\frac{- 72 - (x y) + 9 x + 16 y}{8} - 9 = 0$

Step 4.2.5.1.7.2.3

Factor $- 1$ out of $9 x$ .

$\frac{- 72 - (x y) - (- 9 x) + 16 y}{8} - 9 = 0$

Step 4.2.5.1.7.2.4

Factor $- 1$ out of $16 y$ .

$\frac{- 72 - (x y) - (- 9 x) - (- 16 y)}{8} - 9 = 0$

Step 4.2.5.1.7.2.5

Factor $- 1$ out of $- (x y) - (- 9 x)$ .

$\frac{- 72 - (x y - 9 x) - (- 16 y)}{8} - 9 = 0$

Step 4.2.5.1.7.2.6

Factor $- 1$ out of $- (x y - 9 x) - (- 16 y)$ .

$\frac{- 72 - (x y - 9 x - 16 y)}{8} - 9 = 0$

Step 4.2.5.1.7.3

Factor $- 1$ out of $- 72 - (x y - 9 x - 16 y)$ .

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Step 4.2.5.1.7.3.1

Reorder $- 72$ and $- (x y - 9 x - 16 y)$ .

$\frac{- (x y - 9 x - 16 y) - 72}{8} - 9 = 0$

Step 4.2.5.1.7.3.2

Rewrite $- 72$ as $- 1 (72)$ .

$\frac{- (x y - 9 x - 16 y) - 1 (72)}{8} - 9 = 0$

Step 4.2.5.1.7.3.3

Factor $- 1$ out of $- (x y - 9 x - 16 y) - 1 (72)$ .

$\frac{- (x y - 9 x - 16 y + 72)}{8} - 9 = 0$

Step 4.2.5.2

Move the negative in front of the fraction.

$- \frac{x y - 9 x - 16 y + 72}{8} - 9 = 0$

Step 4.2.6

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- \frac{x y - 9 x - 16 y + 72}{8} - 9 \cdot \frac{8}{8} = 0$

Step 4.2.7

Combine $- 9$ and $\frac{8}{8}$ .

$- \frac{x y - 9 x - 16 y + 72}{8} + \frac{- 9 \cdot 8}{8} = 0$

Step 4.2.8

Combine the numerators over the common denominator.

$\frac{- (x y - 9 x - 16 y + 72) - 9 \cdot 8}{8} = 0$

Step 4.2.9

Simplify the numerator.

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Step 4.2.9.1

Factor $- 1$ out of $- (x y - 9 x - 16 y + 72) - 9 \cdot 8$ .

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Step 4.2.9.1.1

Factor $- 1$ out of $- 9 \cdot 8$ .

$\frac{- (x y - 9 x - 16 y + 72) - (9 \cdot 8)}{8} = 0$

Step 4.2.9.1.2

Factor $- 1$ out of $- (x y - 9 x - 16 y + 72) - (9 \cdot 8)$ .

$\frac{- (x y - 9 x - 16 y + 72 + 9 \cdot 8)}{8} = 0$

Step 4.2.9.2

Multiply $9$ by $8$ .

$\frac{- (x y - 9 x - 16 y + 72 + 72)}{8} = 0$

Step 4.2.9.3

Add $72$ and $72$ .

$\frac{- (x y - 9 x - 16 y + 144)}{8} = 0$

Step 4.2.9.4

Rewrite $x y - 9 x - 16 y + 144$ in a factored form.

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Step 4.2.9.4.1

Factor out the greatest common factor from each group.

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Step 4.2.9.4.1.1

Group the first two terms and the last two terms.

$\frac{- ((x y - 9 x) - 16 y + 144)}{8} = 0$

Step 4.2.9.4.1.2

Factor out the greatest common factor (GCF) from each group.

$\frac{- (x (y - 9) - 16 (y - 9))}{8} = 0$

Step 4.2.9.4.2

Factor the polynomial by factoring out the greatest common factor, $y - 9$ .

$\frac{- ((y - 9) (x - 16))}{8} = 0$

$\frac{- (y - 9) (x - 16)}{8} = 0$

Step 4.2.10

Move the negative in front of the fraction.

$- \frac{(y - 9) (x - 16)}{8} = 0$

Step 4.3

Set the numerator equal to zero.

$(y - 9) (x - 16) = 0$

Step 4.4

Solve the equation for $y$ .

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Step 4.4.1

Divide each term in $(y - 9) (x - 16) = 0$ by $x - 16$ and simplify.

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Step 4.4.1.1

Divide each term in $(y - 9) (x - 16) = 0$ by $x - 16$ .

$\frac{(y - 9) (x - 16)}{x - 16} = \frac{0}{x - 16}$

Step 4.4.1.2

Simplify the left side.

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Step 4.4.1.2.1

Cancel the common factor of $x - 16$ .

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Step 4.4.1.2.1.1

Cancel the common factor.

$\frac{(y - 9) (x - 16)}{x - 16} = \frac{0}{x - 16}$

Step 4.4.1.2.1.2

Divide $y - 9$ by $1$ .

$y - 9 = \frac{0}{x - 16}$

Step 4.4.1.3

Simplify the right side.

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Step 4.4.1.3.1

Divide $0$ by $x - 16$ .

$y - 9 = 0$

Step 4.4.2

Add $9$ to both sides of the equation.

$y = 9$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = 9$

Point-slope form:

$y - 9 = \frac{y - 9}{8} \cdot (x - 8)$

Step 6