Algebra Examples

$f (x) = 2 \frac{1}{2} - 3 \frac{1}{3} x$

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 $x$ .

$0 = 2 \frac{1}{2} - 3 \frac{1}{3} x$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify the right side.

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Step 1.2.1.1

Simplify $2 \frac{1}{2} - 3 \frac{1}{3} x$ .

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Step 1.2.1.1.1

Convert $2 \frac{1}{2}$ to an improper fraction.

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Step 1.2.1.1.1.1

A mixed number is an addition of its whole and fractional parts.

$0 = 2 + \frac{1}{2} - 3 \frac{1}{3} x$

Step 1.2.1.1.1.2

Add $2$ and $\frac{1}{2}$ .

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Step 1.2.1.1.1.2.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.2.1.1.1.2.2

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

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

Step 1.2.1.1.1.2.3

Combine the numerators over the common denominator.

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

Step 1.2.1.1.1.2.4

Simplify the numerator.

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Step 1.2.1.1.1.2.4.1

Multiply $2$ by $2$ .

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

Step 1.2.1.1.1.2.4.2

Add $4$ and $1$ .

$0 = \frac{5}{2} - 3 \frac{1}{3} x$

Step 1.2.1.1.2

Convert $3 \frac{1}{3}$ to an improper fraction.

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Step 1.2.1.1.2.1

A mixed number is an addition of its whole and fractional parts.

$0 = \frac{5}{2} - (3 + \frac{1}{3}) x$

Step 1.2.1.1.2.2

Add $3$ and $\frac{1}{3}$ .

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Step 1.2.1.1.2.2.1

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

$0 = \frac{5}{2} - (3 \cdot \frac{3}{3} + \frac{1}{3}) x$

Step 1.2.1.1.2.2.2

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

$0 = \frac{5}{2} - (\frac{3 \cdot 3}{3} + \frac{1}{3}) x$

Step 1.2.1.1.2.2.3

Combine the numerators over the common denominator.

$0 = \frac{5}{2} - \frac{3 \cdot 3 + 1}{3} x$

Step 1.2.1.1.2.2.4

Simplify the numerator.

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Step 1.2.1.1.2.2.4.1

Multiply $3$ by $3$ .

$0 = \frac{5}{2} - \frac{9 + 1}{3} x$

Step 1.2.1.1.2.2.4.2

Add $9$ and $1$ .

$0 = \frac{5}{2} - \frac{10}{3} x$

Step 1.2.1.1.3

Simplify each term.

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Step 1.2.1.1.3.1

Combine $x$ and $\frac{10}{3}$ .

$0 = \frac{5}{2} - \frac{x \cdot 10}{3}$

Step 1.2.1.1.3.2

Move $10$ to the left of $x$ .

$0 = \frac{5}{2} - \frac{10 x}{3}$

Step 1.2.2

Rewrite the equation as $\frac{5}{2} - \frac{10 x}{3} = 0$ .

$\frac{5}{2} - \frac{10 x}{3} = 0$

Step 1.2.3

Subtract $\frac{5}{2}$ from both sides of the equation.

$- \frac{10 x}{3} = - \frac{5}{2}$

Step 1.2.4

Multiply both sides of the equation by $- \frac{3}{10}$ .

$- \frac{3}{10} (- \frac{10 x}{3}) = - \frac{3}{10} (- \frac{5}{2})$

Step 1.2.5

Simplify both sides of the equation.

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Step 1.2.5.1

Simplify the left side.

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Step 1.2.5.1.1

Simplify $- \frac{3}{10} (- \frac{10 x}{3})$ .

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Step 1.2.5.1.1.1

Cancel the common factor of $3$ .

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Step 1.2.5.1.1.1.1

Move the leading negative in $- \frac{3}{10}$ into the numerator.

$\frac{- 3}{10} (- \frac{10 x}{3}) = - \frac{3}{10} (- \frac{5}{2})$

Step 1.2.5.1.1.1.2

Move the leading negative in $- \frac{10 x}{3}$ into the numerator.

$\frac{- 3}{10} \cdot \frac{- 10 x}{3} = - \frac{3}{10} (- \frac{5}{2})$

Step 1.2.5.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{10} \cdot \frac{- 10 x}{3} = - \frac{3}{10} (- \frac{5}{2})$

Step 1.2.5.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{10} \cdot \frac{- 10 x}{3} = - \frac{3}{10} (- \frac{5}{2})$

Step 1.2.5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{10} (- 10 x) = - \frac{3}{10} (- \frac{5}{2})$

Step 1.2.5.1.1.2

Cancel the common factor of $10$ .

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Step 1.2.5.1.1.2.1

Factor $10$ out of $- 10 x$ .

$\frac{- 1}{10} (10 (- x)) = - \frac{3}{10} (- \frac{5}{2})$

Step 1.2.5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{10} (10 (- x)) = - \frac{3}{10} (- \frac{5}{2})$

Step 1.2.5.1.1.2.3

Rewrite the expression.

$- - x = - \frac{3}{10} (- \frac{5}{2})$

Step 1.2.5.1.1.3

Multiply.

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Step 1.2.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{3}{10} (- \frac{5}{2})$

Step 1.2.5.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{3}{10} (- \frac{5}{2})$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify $- \frac{3}{10} (- \frac{5}{2})$ .

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Step 1.2.5.2.1.1

Cancel the common factor of $5$ .

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Step 1.2.5.2.1.1.1

Move the leading negative in $- \frac{3}{10}$ into the numerator.

$x = \frac{- 3}{10} (- \frac{5}{2})$

Step 1.2.5.2.1.1.2

Move the leading negative in $- \frac{5}{2}$ into the numerator.

$x = \frac{- 3}{10} \cdot \frac{- 5}{2}$

Step 1.2.5.2.1.1.3

Factor $5$ out of $10$ .

$x = \frac{- 3}{5 (2)} \cdot \frac{- 5}{2}$

Step 1.2.5.2.1.1.4

Factor $5$ out of $- 5$ .

$x = \frac{- 3}{5 \cdot 2} \cdot \frac{5 \cdot - 1}{2}$

Step 1.2.5.2.1.1.5

Cancel the common factor.

$x = \frac{- 3}{5 \cdot 2} \cdot \frac{5 \cdot - 1}{2}$

Step 1.2.5.2.1.1.6

Rewrite the expression.

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

Step 1.2.5.2.1.2

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

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

Step 1.2.5.2.1.3

Multiply.

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Step 1.2.5.2.1.3.1

Multiply $- 3$ by $- 1$ .

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

Step 1.2.5.2.1.3.2

Multiply $2$ by $2$ .

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

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{3}{4}, 0)$

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 $y$ .

$y = 2 \frac{1}{2} - 3 \frac{1}{3} \cdot 0$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 \frac{1}{2} - 3 \frac{1}{3} \cdot 0$

Step 2.2.2

Simplify $2 \frac{1}{2} - 3 \frac{1}{3} \cdot 0$ .

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Step 2.2.2.1

Convert $2 \frac{1}{2}$ to an improper fraction.

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Step 2.2.2.1.1

A mixed number is an addition of its whole and fractional parts.

$y = 2 + \frac{1}{2} - 3 \frac{1}{3} \cdot 0$

Step 2.2.2.1.2

Add $2$ and $\frac{1}{2}$ .

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Step 2.2.2.1.2.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = 2 \cdot \frac{2}{2} + \frac{1}{2} - 3 \frac{1}{3} \cdot 0$

Step 2.2.2.1.2.2

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

$y = \frac{2 \cdot 2}{2} + \frac{1}{2} - 3 \frac{1}{3} \cdot 0$

Step 2.2.2.1.2.3

Combine the numerators over the common denominator.

$y = \frac{2 \cdot 2 + 1}{2} - 3 \frac{1}{3} \cdot 0$

Step 2.2.2.1.2.4

Simplify the numerator.

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Step 2.2.2.1.2.4.1

Multiply $2$ by $2$ .

$y = \frac{4 + 1}{2} - 3 \frac{1}{3} \cdot 0$

Step 2.2.2.1.2.4.2

Add $4$ and $1$ .

$y = \frac{5}{2} - 3 \frac{1}{3} \cdot 0$

Step 2.2.2.2

Convert $3 \frac{1}{3}$ to an improper fraction.

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Step 2.2.2.2.1

A mixed number is an addition of its whole and fractional parts.

$y = \frac{5}{2} - (3 + \frac{1}{3}) \cdot 0$

Step 2.2.2.2.2

Add $3$ and $\frac{1}{3}$ .

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Step 2.2.2.2.2.1

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

$y = \frac{5}{2} - (3 \cdot \frac{3}{3} + \frac{1}{3}) \cdot 0$

Step 2.2.2.2.2.2

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

$y = \frac{5}{2} - (\frac{3 \cdot 3}{3} + \frac{1}{3}) \cdot 0$

Step 2.2.2.2.2.3

Combine the numerators over the common denominator.

$y = \frac{5}{2} - \frac{3 \cdot 3 + 1}{3} \cdot 0$

Step 2.2.2.2.2.4

Simplify the numerator.

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Step 2.2.2.2.2.4.1

Multiply $3$ by $3$ .

$y = \frac{5}{2} - \frac{9 + 1}{3} \cdot 0$

Step 2.2.2.2.2.4.2

Add $9$ and $1$ .

$y = \frac{5}{2} - \frac{10}{3} \cdot 0$

Step 2.2.2.3

Multiply $- \frac{10}{3} \cdot 0$ .

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Step 2.2.2.3.1

Multiply $0$ by $- 1$ .

$y = \frac{5}{2} + 0 (\frac{10}{3})$

Step 2.2.2.3.2

Multiply $0$ by $\frac{10}{3}$ .

$y = \frac{5}{2} + 0$

Step 2.2.2.4

Add $\frac{5}{2}$ and $0$ .

$y = \frac{5}{2}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{5}{2})$

Step 3

List the intersections.

x-intercept(s): $(\frac{3}{4}, 0)$

y-intercept(s): $(0, \frac{5}{2})$

Step 4