Algebra Examples

$r (x) = \frac{2 x + 6}{- 6 x + 3}$

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 $x$ .

$0 = \frac{2 x + 6}{- 6 x + 3}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$2 x + 6 = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Subtract $6$ from both sides of the equation.

$2 x = - 6$

Step 1.2.2.2

Divide each term in $2 x = - 6$ by $2$ and simplify.

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Step 1.2.2.2.1

Divide each term in $2 x = - 6$ by $2$ .

$\frac{2 x}{2} = \frac{- 6}{2}$

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 $2$ .

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Step 1.2.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 6}{2}$

Step 1.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{2}$

Step 1.2.2.2.3

Simplify the right side.

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Step 1.2.2.2.3.1

Divide $- 6$ by $2$ .

$x = - 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 3, 0)$

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 $y$ .

$y = \frac{2 (0) + 6}{- 6 \cdot 0 + 3}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{2 (0) + 6}{- 6 \cdot 0 + 3}$

Step 2.2.2

Simplify $\frac{2 (0) + 6}{- 6 \cdot 0 + 3}$ .

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Step 2.2.2.1

Cancel the common factor of $2 (0) + 6$ and $- 6 \cdot 0 + 3$ .

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Step 2.2.2.1.1

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

$y = \frac{3 (2 \cdot (0)) + 6}{- 6 \cdot 0 + 3}$

Step 2.2.2.1.2

Factor $3$ out of $6$ .

$y = \frac{3 (2 \cdot (0)) + 3 (2)}{- 6 \cdot 0 + 3}$

Step 2.2.2.1.3

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

$y = \frac{3 (2 \cdot (0) + 2)}{- 6 \cdot 0 + 3}$

Step 2.2.2.1.4

Cancel the common factors.

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Step 2.2.2.1.4.1

Factor $3$ out of $- 6 \cdot 0$ .

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

Step 2.2.2.1.4.2

Factor $3$ out of $3$ .

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

Step 2.2.2.1.4.3

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

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

Step 2.2.2.1.4.4

Cancel the common factor.

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

Step 2.2.2.1.4.5

Rewrite the expression.

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

Step 2.2.2.2

Simplify the numerator.

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Step 2.2.2.2.1

Multiply $2$ by $0$ .

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

Step 2.2.2.2.2

Add $0$ and $2$ .

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

Step 2.2.2.3

Simplify the denominator.

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Step 2.2.2.3.1

Multiply $- 2$ 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): $(- 3, 0)$

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

Step 4