Trigonometry Examples

$12 x - 7 y = 27$

Step 1

Solve for $y$ .

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Step 1.1

Subtract $12 x$ from both sides of the equation.

$- 7 y = 27 - 12 x$

Step 1.2

Divide each term in $- 7 y = 27 - 12 x$ by $- 7$ and simplify.

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Step 1.2.1

Divide each term in $- 7 y = 27 - 12 x$ by $- 7$ .

$\frac{- 7 y}{- 7} = \frac{27}{- 7} + \frac{- 12 x}{- 7}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 7$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 7 y}{- 7} = \frac{27}{- 7} + \frac{- 12 x}{- 7}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{27}{- 7} + \frac{- 12 x}{- 7}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{27}{7} + \frac{- 12 x}{- 7}$

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

$y = - \frac{27}{7} + \frac{12 x}{7}$

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 $- \frac{27}{7}$ and $\frac{12 x}{7}$ .

$y = \frac{12 x}{7} - \frac{27}{7}$

Step 2.3

Reorder terms.

$y = \frac{12}{7} x - \frac{27}{7}$

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 $y = m x + b$ .

$m = \frac{12}{7}$

$b = - \frac{27}{7}$

Step 3.2

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

Slope: $\frac{12}{7}$

y-intercept: $(0, - \frac{27}{7})$

Slope: $\frac{12}{7}$

y-intercept: $(0, - \frac{27}{7})$

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 $- \frac{27}{7}$ and $\frac{12 x}{7}$ .

$y = \frac{12 x}{7} - \frac{27}{7}$

Step 4.1.2

Reorder terms.

$y = \frac{12}{7} x - \frac{27}{7}$

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 $x$ .

$0 = \frac{12}{7} x - \frac{27}{7}$

Step 4.2.2

Solve the equation.

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Step 4.2.2.1

Rewrite the equation as $\frac{12}{7} x - \frac{27}{7} = 0$ .

$\frac{12}{7} x - \frac{27}{7} = 0$

Step 4.2.2.2

Combine $\frac{12}{7}$ and $x$ .

$\frac{12 x}{7} - \frac{27}{7} = 0$

Step 4.2.2.3

Add $\frac{27}{7}$ to both sides of the equation.

$\frac{12 x}{7} = \frac{27}{7}$

Step 4.2.2.4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$12 x = 27$

Step 4.2.2.5

Divide each term in $12 x = 27$ by $12$ and simplify.

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Step 4.2.2.5.1

Divide each term in $12 x = 27$ by $12$ .

$\frac{12 x}{12} = \frac{27}{12}$

Step 4.2.2.5.2

Simplify the left side.

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Step 4.2.2.5.2.1

Cancel the common factor of $12$ .

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Step 4.2.2.5.2.1.1

Cancel the common factor.

$\frac{12 x}{12} = \frac{27}{12}$

Step 4.2.2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{27}{12}$

Step 4.2.2.5.3

Simplify the right side.

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Step 4.2.2.5.3.1

Cancel the common factor of $27$ and $12$ .

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Step 4.2.2.5.3.1.1

Factor $3$ out of $27$ .

$x = \frac{3 (9)}{12}$

Step 4.2.2.5.3.1.2

Cancel the common factors.

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Step 4.2.2.5.3.1.2.1

Factor $3$ out of $12$ .

$x = \frac{3 \cdot 9}{3 \cdot 4}$

Step 4.2.2.5.3.1.2.2

Cancel the common factor.

$x = \frac{3 \cdot 9}{3 \cdot 4}$

Step 4.2.2.5.3.1.2.3

Rewrite the expression.

$x = \frac{9}{4}$

Step 4.2.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{9}{4}, 0)$

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 $y$ .

$y = \frac{12}{7} \cdot (0) - \frac{27}{7}$

Step 4.3.2

Solve the equation.

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Step 4.3.2.1

Multiply $\frac{12}{7}$ by $0$ .

$y = \frac{12}{7} \cdot 0 - \frac{27}{7}$

Step 4.3.2.2

Remove parentheses.

$y = \frac{12}{7} \cdot (0) - \frac{27}{7}$

Step 4.3.2.3

Simplify $\frac{12}{7} \cdot (0) - \frac{27}{7}$ .

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Step 4.3.2.3.1

Multiply $\frac{12}{7}$ by $0$ .

$y = 0 - \frac{27}{7}$

Step 4.3.2.3.2

Subtract $\frac{27}{7}$ from $0$ .

$y = - \frac{27}{7}$

Step 4.3.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{27}{7})$

Step 4.4

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

$\begin{array}{c} x & y \\ 0 & - \frac{27}{7} \\ \frac{9}{4} & 0 \end{array}$

Step 5

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

Slope: $\frac{12}{7}$

y-intercept: $(0, - \frac{27}{7})$

$\begin{array}{c} x & y \\ 0 & - \frac{27}{7} \\ \frac{9}{4} & 0 \end{array}$

Step 6