Precalculus Examples

$y - 1 = \frac{3}{4} \cdot (x - 6)$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$(0) - 1 = \frac{3}{4} \cdot (x - 6)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{3}{4} \cdot (x - 6) = (0) - 1$ .

$\frac{3}{4} \cdot (x - 6) = (0) - 1$

Step 1.2.2

Multiply both sides of the equation by $\frac{4}{3}$ .

$\frac{4}{3} (\frac{3}{4} \cdot (x - 6)) = \frac{4}{3} ((0) - 1)$

Step 1.2.3

Simplify both sides of the equation.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Simplify $\frac{4}{3} (\frac{3}{4} \cdot (x - 6))$ .

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Step 1.2.3.1.1.1

Apply the distributive property.

$\frac{4}{3} (\frac{3}{4} x + \frac{3}{4} \cdot - 6) = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.2

Combine $\frac{3}{4}$ and $x$ .

$\frac{4}{3} (\frac{3 x}{4} + \frac{3}{4} \cdot - 6) = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.3

Cancel the common factor of $2$ .

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Step 1.2.3.1.1.3.1

Factor $2$ out of $4$ .

$\frac{4}{3} (\frac{3 x}{4} + \frac{3}{2 (2)} \cdot - 6) = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.3.2

Factor $2$ out of $- 6$ .

$\frac{4}{3} (\frac{3 x}{4} + \frac{3}{2 \cdot 2} \cdot (2 \cdot - 3)) = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.3.3

Cancel the common factor.

$\frac{4}{3} (\frac{3 x}{4} + \frac{3}{2 \cdot 2} \cdot (2 \cdot - 3)) = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.3.4

Rewrite the expression.

$\frac{4}{3} (\frac{3 x}{4} + \frac{3}{2} \cdot - 3) = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.4

Combine $\frac{3}{2}$ and $- 3$ .

$\frac{4}{3} (\frac{3 x}{4} + \frac{3 \cdot - 3}{2}) = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.5

Simplify the expression.

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Step 1.2.3.1.1.5.1

Multiply $3$ by $- 3$ .

$\frac{4}{3} (\frac{3 x}{4} + \frac{- 9}{2}) = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.5.2

Move the negative in front of the fraction.

$\frac{4}{3} (\frac{3 x}{4} - \frac{9}{2}) = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.6

Apply the distributive property.

$\frac{4}{3} \cdot \frac{3 x}{4} + \frac{4}{3} (- \frac{9}{2}) = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.7

Cancel the common factor of $4$ .

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Step 1.2.3.1.1.7.1

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 x}{4} + \frac{4}{3} (- \frac{9}{2}) = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.7.2

Rewrite the expression.

$\frac{1}{3} (3 x) + \frac{4}{3} (- \frac{9}{2}) = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.8

Cancel the common factor of $3$ .

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Step 1.2.3.1.1.8.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) + \frac{4}{3} (- \frac{9}{2}) = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.8.2

Cancel the common factor.

$\frac{1}{3} (3 x) + \frac{4}{3} (- \frac{9}{2}) = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.8.3

Rewrite the expression.

$x + \frac{4}{3} (- \frac{9}{2}) = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.9

Cancel the common factor of $2$ .

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Step 1.2.3.1.1.9.1

Move the leading negative in $- \frac{9}{2}$ into the numerator.

$x + \frac{4}{3} \cdot \frac{- 9}{2} = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.9.2

Factor $2$ out of $4$ .

$x + \frac{2 (2)}{3} \cdot \frac{- 9}{2} = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.9.3

Cancel the common factor.

$x + \frac{2 \cdot 2}{3} \cdot \frac{- 9}{2} = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.9.4

Rewrite the expression.

$x + \frac{2}{3} \cdot - 9 = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.10

Cancel the common factor of $3$ .

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Step 1.2.3.1.1.10.1

Factor $3$ out of $- 9$ .

$x + \frac{2}{3} \cdot (3 (- 3)) = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.10.2

Cancel the common factor.

$x + \frac{2}{3} \cdot (3 \cdot - 3) = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.10.3

Rewrite the expression.

$x + 2 \cdot - 3 = \frac{4}{3} ((0) - 1)$

Step 1.2.3.1.1.11

Multiply $2$ by $- 3$ .

$x - 6 = \frac{4}{3} ((0) - 1)$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Simplify $\frac{4}{3} ((0) - 1)$ .

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Step 1.2.3.2.1.1

Subtract $1$ from $0$ .

$x - 6 = \frac{4}{3} \cdot - 1$

Step 1.2.3.2.1.2

Multiply $\frac{4}{3} \cdot - 1$ .

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Step 1.2.3.2.1.2.1

Combine $\frac{4}{3}$ and $- 1$ .

$x - 6 = \frac{4 \cdot - 1}{3}$

Step 1.2.3.2.1.2.2

Multiply $4$ by $- 1$ .

$x - 6 = \frac{- 4}{3}$

Step 1.2.3.2.1.3

Move the negative in front of the fraction.

$x - 6 = - \frac{4}{3}$

Step 1.2.4

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

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Step 1.2.4.1

Add $6$ to both sides of the equation.

$x = - \frac{4}{3} + 6$

Step 1.2.4.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = - \frac{4}{3} + 6 \cdot \frac{3}{3}$

Step 1.2.4.3

Combine $6$ and $\frac{3}{3}$ .

$x = - \frac{4}{3} + \frac{6 \cdot 3}{3}$

Step 1.2.4.4

Combine the numerators over the common denominator.

$x = \frac{- 4 + 6 \cdot 3}{3}$

Step 1.2.4.5

Simplify the numerator.

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Step 1.2.4.5.1

Multiply $6$ by $3$ .

$x = \frac{- 4 + 18}{3}$

Step 1.2.4.5.2

Add $- 4$ and $18$ .

$x = \frac{14}{3}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{14}{3}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y - 1 = \frac{3}{4} \cdot ((0) - 6)$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify $\frac{3}{4} \cdot ((0) - 6)$ .

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Step 2.2.1.1

Subtract $6$ from $0$ .

$y - 1 = \frac{3}{4} \cdot - 6$

Step 2.2.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.2.1

Factor $2$ out of $4$ .

$y - 1 = \frac{3}{2 (2)} \cdot - 6$

Step 2.2.1.2.2

Factor $2$ out of $- 6$ .

$y - 1 = \frac{3}{2 \cdot 2} \cdot (2 \cdot - 3)$

Step 2.2.1.2.3

Cancel the common factor.

$y - 1 = \frac{3}{2 \cdot 2} \cdot (2 \cdot - 3)$

Step 2.2.1.2.4

Rewrite the expression.

$y - 1 = \frac{3}{2} \cdot - 3$

Step 2.2.1.3

Combine $\frac{3}{2}$ and $- 3$ .

$y - 1 = \frac{3 \cdot - 3}{2}$

Step 2.2.1.4

Simplify the expression.

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Step 2.2.1.4.1

Multiply $3$ by $- 3$ .

$y - 1 = \frac{- 9}{2}$

Step 2.2.1.4.2

Move the negative in front of the fraction.

$y - 1 = - \frac{9}{2}$

Step 2.2.2

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

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Step 2.2.2.1

Add $1$ to both sides of the equation.

$y = - \frac{9}{2} + 1$

Step 2.2.2.2

Write $1$ as a fraction with a common denominator.

$y = - \frac{9}{2} + \frac{2}{2}$

Step 2.2.2.3

Combine the numerators over the common denominator.

$y = \frac{- 9 + 2}{2}$

Step 2.2.2.4

Add $- 9$ and $2$ .

$y = \frac{- 7}{2}$

Step 2.2.2.5

Move the negative in front of the fraction.

$y = - \frac{7}{2}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{7}{2})$

Step 3

List the intersections.

x-intercept(s): $(\frac{14}{3}, 0)$

y-intercept(s): $(0, - \frac{7}{2})$

Step 4