Trigonometry Examples

$- 5 x + 6 y = 2$

Step 1

Solve for $y$ .

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Step 1.1

Add $5 x$ to both sides of the equation.

$6 y = 2 + 5 x$

Step 1.2

Divide each term in $6 y = 2 + 5 x$ by $6$ and simplify.

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Step 1.2.1

Divide each term in $6 y = 2 + 5 x$ by $6$ .

$\frac{6 y}{6} = \frac{2}{6} + \frac{5 x}{6}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $6$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{2}{6} + \frac{5 x}{6}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{2}{6} + \frac{5 x}{6}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $2$ and $6$ .

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Step 1.2.3.1.1

Factor $2$ out of $2$ .

$y = \frac{2 (1)}{6} + \frac{5 x}{6}$

Step 1.2.3.1.2

Cancel the common factors.

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Step 1.2.3.1.2.1

Factor $2$ out of $6$ .

$y = \frac{2 \cdot 1}{2 \cdot 3} + \frac{5 x}{6}$

Step 1.2.3.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot 1}{2 \cdot 3} + \frac{5 x}{6}$

Step 1.2.3.1.2.3

Rewrite the expression.

$y = \frac{1}{3} + \frac{5 x}{6}$

Step 2

Rewrite in slope-intercept form.

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Step 2.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 2.2

Reorder $\frac{1}{3}$ and $\frac{5 x}{6}$ .

$y = \frac{5 x}{6} + \frac{1}{3}$

Step 2.3

Reorder terms.

$y = \frac{5}{6} x + \frac{1}{3}$

Step 3

Use the slope-intercept form to find the slope and y-intercept.

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Step 3.1

Find the values of $m$ and $b$ using the form $y = m x + b$ .

$m = \frac{5}{6}$

$b = \frac{1}{3}$

Step 3.2

The slope of the line is the value of $m$ , and the y-intercept is the value of $b$ .

Slope: $\frac{5}{6}$

y-intercept: $(0, \frac{1}{3})$

Slope: $\frac{5}{6}$

y-intercept: $(0, \frac{1}{3})$

Step 4

Any line can be graphed using two points. Select two $x$ values, and plug them into the equation to find the corresponding $y$ values.

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Step 4.1

Write in $y = m x + b$ form.

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Step 4.1.1

Reorder $\frac{1}{3}$ and $\frac{5 x}{6}$ .

$y = \frac{5 x}{6} + \frac{1}{3}$

Step 4.1.2

Reorder terms.

$y = \frac{5}{6} x + \frac{1}{3}$

Step 4.2

Find the x-intercept.

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Step 4.2.1

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

$0 = \frac{5}{6} x + \frac{1}{3}$

Step 4.2.2

Solve the equation.

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Step 4.2.2.1

Rewrite the equation as $\frac{5}{6} x + \frac{1}{3} = 0$ .

$\frac{5}{6} x + \frac{1}{3} = 0$

Step 4.2.2.2

Combine $\frac{5}{6}$ and $x$ .

$\frac{5 x}{6} + \frac{1}{3} = 0$

Step 4.2.2.3

Subtract $\frac{1}{3}$ from both sides of the equation.

$\frac{5 x}{6} = - \frac{1}{3}$

Step 4.2.2.4

Multiply both sides of the equation by $\frac{6}{5}$ .

$\frac{6}{5} \cdot \frac{5 x}{6} = \frac{6}{5} (- \frac{1}{3})$

Step 4.2.2.5

Simplify both sides of the equation.

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Step 4.2.2.5.1

Simplify the left side.

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Step 4.2.2.5.1.1

Simplify $\frac{6}{5} \cdot \frac{5 x}{6}$ .

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Step 4.2.2.5.1.1.1

Cancel the common factor of $6$ .

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Step 4.2.2.5.1.1.1.1

Cancel the common factor.

$\frac{6}{5} \cdot \frac{5 x}{6} = \frac{6}{5} (- \frac{1}{3})$

Step 4.2.2.5.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (5 x) = \frac{6}{5} (- \frac{1}{3})$

Step 4.2.2.5.1.1.2

Cancel the common factor of $5$ .

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Step 4.2.2.5.1.1.2.1

Factor $5$ out of $5 x$ .

$\frac{1}{5} (5 (x)) = \frac{6}{5} (- \frac{1}{3})$

Step 4.2.2.5.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 x) = \frac{6}{5} (- \frac{1}{3})$

Step 4.2.2.5.1.1.2.3

Rewrite the expression.

$x = \frac{6}{5} (- \frac{1}{3})$

Step 4.2.2.5.2

Simplify the right side.

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Step 4.2.2.5.2.1

Simplify $\frac{6}{5} (- \frac{1}{3})$ .

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Step 4.2.2.5.2.1.1

Cancel the common factor of $3$ .

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Step 4.2.2.5.2.1.1.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$x = \frac{6}{5} \cdot \frac{- 1}{3}$

Step 4.2.2.5.2.1.1.2

Factor $3$ out of $6$ .

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

Step 4.2.2.5.2.1.1.3

Cancel the common factor.

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

Step 4.2.2.5.2.1.1.4

Rewrite the expression.

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

Step 4.2.2.5.2.1.2

Combine $\frac{2}{5}$ and $- 1$ .

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

Step 4.2.2.5.2.1.3

Simplify the expression.

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Step 4.2.2.5.2.1.3.1

Multiply $2$ by $- 1$ .

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

Step 4.2.2.5.2.1.3.2

Move the negative in front of the fraction.

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

Step 4.2.3

x-intercept(s) in point form.

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

Step 4.3

Find the y-intercept.

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Step 4.3.1

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

$y = \frac{5}{6} \cdot (0) + \frac{1}{3}$

Step 4.3.2

Solve the equation.

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Step 4.3.2.1

Multiply $\frac{5}{6}$ by $0$ .

$y = \frac{5}{6} \cdot 0 + \frac{1}{3}$

Step 4.3.2.2

Remove parentheses.

$y = \frac{5}{6} \cdot (0) + \frac{1}{3}$

Step 4.3.2.3

Simplify $\frac{5}{6} \cdot (0) + \frac{1}{3}$ .

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Step 4.3.2.3.1

Multiply $\frac{5}{6}$ by $0$ .

$y = 0 + \frac{1}{3}$

Step 4.3.2.3.2

Add $0$ and $\frac{1}{3}$ .

$y = \frac{1}{3}$

Step 4.3.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{1}{3})$

Step 4.4

Create a table of the $x$ and $y$ values.

$\begin{array}{c} x & y \\ - \frac{2}{5} & 0 \\ 0 & \frac{1}{3} \end{array}$

Step 5

Graph the line using the slope and the y-intercept, or the points.

Slope: $\frac{5}{6}$

y-intercept: $(0, \frac{1}{3})$

$\begin{array}{c} x & y \\ - \frac{2}{5} & 0 \\ 0 & \frac{1}{3} \end{array}$

Step 6