Precalculus Examples

$f (x) = \frac{5 x - 4}{3 x + 2}$

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 $x$ .

$0 = \frac{5 x - 4}{3 x + 2}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$5 x - 4 = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Add $4$ to both sides of the equation.

$5 x = 4$

Step 1.2.2.2

Divide each term in $5 x = 4$ by $5$ and simplify.

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Step 1.2.2.2.1

Divide each term in $5 x = 4$ by $5$ .

$\frac{5 x}{5} = \frac{4}{5}$

Step 1.2.2.2.2

Simplify the left side.

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Step 1.2.2.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{4}{5}$

Step 1.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{5}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{4}{5}, 0)$

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 $y$ .

$y = \frac{5 (0) - 4}{3 (0) + 2}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{5 (0) - 4}{3 (0) + 2}$

Step 2.2.2

Simplify $\frac{5 (0) - 4}{3 (0) + 2}$ .

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Step 2.2.2.1

Cancel the common factor of $5 (0) - 4$ and $3 (0) + 2$ .

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Step 2.2.2.1.1

Factor $2$ out of $5 (0)$ .

$y = \frac{2 (5 \cdot (0)) - 4}{3 (0) + 2}$

Step 2.2.2.1.2

Factor $2$ out of $- 4$ .

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

Step 2.2.2.1.3

Factor $2$ out of $2 (5 \cdot (0)) + 2 (- 2)$ .

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

Step 2.2.2.1.4

Cancel the common factors.

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Step 2.2.2.1.4.1

Factor $2$ out of $3 (0)$ .

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

Step 2.2.2.1.4.2

Factor $2$ out of $2$ .

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

Step 2.2.2.1.4.3

Factor $2$ out of $2 (3 \cdot (0)) + 2 (1)$ .

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

Step 2.2.2.1.4.4

Cancel the common factor.

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

Step 2.2.2.1.4.5

Rewrite the expression.

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

Step 2.2.2.2

Simplify the numerator.

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Step 2.2.2.2.1

Multiply $5$ by $0$ .

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

Step 2.2.2.2.2

Subtract $2$ from $0$ .

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

Step 2.2.2.3

Simplify the denominator.

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Step 2.2.2.3.1

Multiply $3$ by $0$ .

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

Step 2.2.2.3.2

Add $0$ and $1$ .

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

Step 2.2.2.4

Divide $- 2$ by $1$ .

$y = - 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 2)$

Step 3

List the intersections.

x-intercept(s): $(\frac{4}{5}, 0)$

y-intercept(s): $(0, - 2)$

Step 4