Algebra Examples

$(- 3, 5)$ , $(4, 6)$

Step 1

Find the slope of the line between $(- 3, 5)$ and $(4, 6)$ 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.

$m = \frac{6 - (5)}{4 - (- 3)}$

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 $5$ .

$m = \frac{6 - 5}{4 - (- 3)}$

Step 1.4.1.2

Subtract $5$ from $6$ .

$m = \frac{1}{4 - (- 3)}$

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Multiply $- 1$ by $- 3$ .

$m = \frac{1}{4 + 3}$

Step 1.4.2.2

Add $4$ and $3$ .

$m = \frac{1}{7}$

Step 2

Use the slope $\frac{1}{7}$ and a given point $(- 3, 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = \frac{1}{7} \cdot (x - (- 3))$

Step 3

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

$y - 5 = \frac{1}{7} \cdot (x + 3)$

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 $7$ .

$7 (y - 5) = 7 (\frac{1}{7} \cdot (x + 3))$

Step 4.3

Simplify the left side.

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Step 4.3.1

Simplify $7 (y - 5)$ .

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Step 4.3.1.1

Apply the distributive property.

$7 y + 7 \cdot - 5 = 7 (\frac{1}{7} \cdot (x + 3))$

Step 4.3.1.2

Multiply $7$ by $- 5$ .

$7 y - 35 = 7 (\frac{1}{7} \cdot (x + 3))$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $7 (\frac{1}{7} \cdot (x + 3))$ .

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Step 4.4.1.1

Apply the distributive property.

$7 y - 35 = 7 (\frac{1}{7} x + \frac{1}{7} \cdot 3)$

Step 4.4.1.2

Combine $\frac{1}{7}$ and $x$ .

$7 y - 35 = 7 (\frac{x}{7} + \frac{1}{7} \cdot 3)$

Step 4.4.1.3

Combine $\frac{1}{7}$ and $3$ .

$7 y - 35 = 7 (\frac{x}{7} + \frac{3}{7})$

Step 4.4.1.4

Apply the distributive property.

$7 y - 35 = 7 \frac{x}{7} + 7 (\frac{3}{7})$

Step 4.4.1.5

Cancel the common factor of $7$ .

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Step 4.4.1.5.1

Cancel the common factor.

$7 y - 35 = 7 \frac{x}{7} + 7 (\frac{3}{7})$

Step 4.4.1.5.2

Rewrite the expression.

$7 y - 35 = x + 7 (\frac{3}{7})$

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.

$7 y - 35 = x + 7 (\frac{3}{7})$

Step 4.4.1.6.2

Rewrite the expression.

$7 y - 35 = x + 3$

Step 4.5

Rewrite the equation.

$x + 3 = 7 y - 35$

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.

$x + 3 - 7 y = - 35$

Step 4.6.2

Move $3$ .

$x - 7 y + 3 = - 35$

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

Subtract $3$ from both sides of the equation.

$x - 7 y = - 35 - 3$

Step 4.7.2

Subtract $3$ from $- 35$ .

$x - 7 y = - 38$

Step 5