Algebra Examples

$2 x - \frac{2}{3} = \frac{3}{4} y + 3$

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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Step 1.2.1.1

Multiply $\frac{3}{4}$ by $0$ .

$2 x - \frac{2}{3} = 0 + 3$

Step 1.2.1.2

Add $0$ and $3$ .

$2 x - \frac{2}{3} = 3$

Step 1.2.2

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

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Step 1.2.2.1

Add $\frac{2}{3}$ to both sides of the equation.

$2 x = 3 + \frac{2}{3}$

Step 1.2.2.2

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

$2 x = 3 \cdot \frac{3}{3} + \frac{2}{3}$

Step 1.2.2.3

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

$2 x = \frac{3 \cdot 3}{3} + \frac{2}{3}$

Step 1.2.2.4

Combine the numerators over the common denominator.

$2 x = \frac{3 \cdot 3 + 2}{3}$

Step 1.2.2.5

Simplify the numerator.

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Step 1.2.2.5.1

Multiply $3$ by $3$ .

$2 x = \frac{9 + 2}{3}$

Step 1.2.2.5.2

Add $9$ and $2$ .

$2 x = \frac{11}{3}$

Step 1.2.3

Divide each term in $2 x = \frac{11}{3}$ by $2$ and simplify.

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Step 1.2.3.1

Divide each term in $2 x = \frac{11}{3}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{11}{3}}{2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{11}{3}}{2}$

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{11}{3}}{2}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{11}{3} \cdot \frac{1}{2}$

Step 1.2.3.3.2

Multiply $\frac{11}{3} \cdot \frac{1}{2}$ .

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Step 1.2.3.3.2.1

Multiply $\frac{11}{3}$ by $\frac{1}{2}$ .

$x = \frac{11}{3 \cdot 2}$

Step 1.2.3.3.2.2

Multiply $3$ by $2$ .

$x = \frac{11}{6}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{11}{6}, 0)$

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 $y$ .

$2 (0) - \frac{2}{3} = \frac{3}{4} y + 3$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $\frac{3}{4} y + 3 = 2 (0) - \frac{2}{3}$ .

$\frac{3}{4} y + 3 = 2 (0) - \frac{2}{3}$

Step 2.2.2

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

$\frac{3 y}{4} + 3 = 2 (0) - \frac{2}{3}$

Step 2.2.3

Simplify $2 (0) - \frac{2}{3}$ .

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Step 2.2.3.1

Multiply $2$ by $0$ .

$\frac{3 y}{4} + 3 = 0 - \frac{2}{3}$

Step 2.2.3.2

Subtract $\frac{2}{3}$ from $0$ .

$\frac{3 y}{4} + 3 = - \frac{2}{3}$

Step 2.2.4

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

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Step 2.2.4.1

Subtract $3$ from both sides of the equation.

$\frac{3 y}{4} = - \frac{2}{3} - 3$

Step 2.2.4.2

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

$\frac{3 y}{4} = - \frac{2}{3} - 3 \cdot \frac{3}{3}$

Step 2.2.4.3

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

$\frac{3 y}{4} = - \frac{2}{3} + \frac{- 3 \cdot 3}{3}$

Step 2.2.4.4

Combine the numerators over the common denominator.

$\frac{3 y}{4} = \frac{- 2 - 3 \cdot 3}{3}$

Step 2.2.4.5

Simplify the numerator.

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Step 2.2.4.5.1

Multiply $- 3$ by $3$ .

$\frac{3 y}{4} = \frac{- 2 - 9}{3}$

Step 2.2.4.5.2

Subtract $9$ from $- 2$ .

$\frac{3 y}{4} = \frac{- 11}{3}$

Step 2.2.4.6

Move the negative in front of the fraction.

$\frac{3 y}{4} = - \frac{11}{3}$

Step 2.2.5

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

$\frac{4}{3} \cdot \frac{3 y}{4} = \frac{4}{3} (- \frac{11}{3})$

Step 2.2.6

Simplify both sides of the equation.

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Step 2.2.6.1

Simplify the left side.

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Step 2.2.6.1.1

Simplify $\frac{4}{3} \cdot \frac{3 y}{4}$ .

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Step 2.2.6.1.1.1

Cancel the common factor of $4$ .

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Step 2.2.6.1.1.1.1

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 y}{4} = \frac{4}{3} (- \frac{11}{3})$

Step 2.2.6.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 y) = \frac{4}{3} (- \frac{11}{3})$

Step 2.2.6.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.6.1.1.2.1

Factor $3$ out of $3 y$ .

$\frac{1}{3} (3 (y)) = \frac{4}{3} (- \frac{11}{3})$

Step 2.2.6.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 y) = \frac{4}{3} (- \frac{11}{3})$

Step 2.2.6.1.1.2.3

Rewrite the expression.

$y = \frac{4}{3} (- \frac{11}{3})$

Step 2.2.6.2

Simplify the right side.

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Step 2.2.6.2.1

Multiply $\frac{4}{3} (- \frac{11}{3})$ .

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Step 2.2.6.2.1.1

Multiply $\frac{4}{3}$ by $\frac{11}{3}$ .

$y = - \frac{4 \cdot 11}{3 \cdot 3}$

Step 2.2.6.2.1.2

Multiply $4$ by $11$ .

$y = - \frac{44}{3 \cdot 3}$

Step 2.2.6.2.1.3

Multiply $3$ by $3$ .

$y = - \frac{44}{9}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{44}{9})$

Step 3

List the intersections.

x-intercept(s): $(\frac{11}{6}, 0)$

y-intercept(s): $(0, - \frac{44}{9})$

Step 4