Algebra Examples

$(2, 7)$ and $(- 5, 2)$

Step 1

Find the slope of the line between $(2, 7)$ and $(- 5, 2)$ 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.

$m = \frac{2 - (7)}{- 5 - (2)}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Multiply $- 1$ by $7$ .

$m = \frac{2 - 7}{- 5 - (2)}$

Step 1.4.1.2

Subtract $7$ from $2$ .

$m = \frac{- 5}{- 5 - (2)}$

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Multiply $- 1$ by $2$ .

$m = \frac{- 5}{- 5 - 2}$

Step 1.4.2.2

Subtract $2$ from $- 5$ .

$m = \frac{- 5}{- 7}$

Step 1.4.3

Dividing two negative values results in a positive value.

$m = \frac{5}{7}$

Step 2

Use the slope $\frac{5}{7}$ and a given point $(2, 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (7) = \frac{5}{7} \cdot (x - (2))$

Step 3

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

$y - 7 = \frac{5}{7} \cdot (x - 2)$

Step 4

Write the equation in standard form.

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Step 4.1

The standard form of a linear equation is $A x + B y = C$ .

Step 4.2

Multiply both sides by $7$ .

$7 (y - 7) = 7 (\frac{5}{7} \cdot (x - 2))$

Step 4.3

Simplify the left side.

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Step 4.3.1

Simplify $7 (y - 7)$ .

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Step 4.3.1.1

Apply the distributive property.

$7 y + 7 \cdot - 7 = 7 (\frac{5}{7} \cdot (x - 2))$

Step 4.3.1.2

Multiply $7$ by $- 7$ .

$7 y - 49 = 7 (\frac{5}{7} \cdot (x - 2))$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $7 (\frac{5}{7} \cdot (x - 2))$ .

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Step 4.4.1.1

Apply the distributive property.

$7 y - 49 = 7 (\frac{5}{7} x + \frac{5}{7} \cdot - 2)$

Step 4.4.1.2

Combine $\frac{5}{7}$ and $x$ .

$7 y - 49 = 7 (\frac{5 x}{7} + \frac{5}{7} \cdot - 2)$

Step 4.4.1.3

Multiply $\frac{5}{7} \cdot - 2$ .

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Step 4.4.1.3.1

Combine $\frac{5}{7}$ and $- 2$ .

$7 y - 49 = 7 (\frac{5 x}{7} + \frac{5 \cdot - 2}{7})$

Step 4.4.1.3.2

Multiply $5$ by $- 2$ .

$7 y - 49 = 7 (\frac{5 x}{7} + \frac{- 10}{7})$

Step 4.4.1.4

Move the negative in front of the fraction.

$7 y - 49 = 7 (\frac{5 x}{7} - \frac{10}{7})$

Step 4.4.1.5

Apply the distributive property.

$7 y - 49 = 7 \frac{5 x}{7} + 7 (- \frac{10}{7})$

Step 4.4.1.6

Cancel the common factor of $7$ .

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Step 4.4.1.6.1

Cancel the common factor.

$7 y - 49 = 7 \frac{5 x}{7} + 7 (- \frac{10}{7})$

Step 4.4.1.6.2

Rewrite the expression.

$7 y - 49 = 5 x + 7 (- \frac{10}{7})$

Step 4.4.1.7

Cancel the common factor of $7$ .

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Step 4.4.1.7.1

Move the leading negative in $- \frac{10}{7}$ into the numerator.

$7 y - 49 = 5 x + 7 (\frac{- 10}{7})$

Step 4.4.1.7.2

Cancel the common factor.

$7 y - 49 = 5 x + 7 (\frac{- 10}{7})$

Step 4.4.1.7.3

Rewrite the expression.

$7 y - 49 = 5 x - 10$

Step 4.5

Rewrite the equation.

$5 x - 10 = 7 y - 49$

Step 4.6

Move all terms containing variables to the left side of the equation.

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Step 4.6.1

Subtract $7 y$ from both sides of the equation.

$5 x - 10 - 7 y = - 49$

Step 4.6.2

Move $- 10$ .

$5 x - 7 y - 10 = - 49$

Step 4.7

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

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Step 4.7.1

Add $10$ to both sides of the equation.

$5 x - 7 y = - 49 + 10$

Step 4.7.2

Add $- 49$ and $10$ .

$5 x - 7 y = - 39$

Step 5