Algebra Examples

$7 x + 3 y \leq - 24$ $x + 3 y \leq - 6$

Step 1

Graph $7 x + 3 y \leq - 24$ .

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Step 1.1

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

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Step 1.1.1

Solve for $y$ .

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Step 1.1.1.1

Subtract $7 x$ from both sides of the inequality.

$3 y \leq - 24 - 7 x$

Step 1.1.1.2

Divide each term in $3 y \leq - 24 - 7 x$ by $3$ and simplify.

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Step 1.1.1.2.1

Divide each term in $3 y \leq - 24 - 7 x$ by $3$ .

$\frac{3 y}{3} \leq \frac{- 24}{3} + \frac{- 7 x}{3}$

Step 1.1.1.2.2

Simplify the left side.

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Step 1.1.1.2.2.1

Cancel the common factor of $3$ .

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Step 1.1.1.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} \leq \frac{- 24}{3} + \frac{- 7 x}{3}$

Step 1.1.1.2.2.1.2

Divide $y$ by $1$ .

$y \leq \frac{- 24}{3} + \frac{- 7 x}{3}$

Step 1.1.1.2.3

Simplify the right side.

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Step 1.1.1.2.3.1

Simplify each term.

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Step 1.1.1.2.3.1.1

Divide $- 24$ by $3$ .

$y \leq - 8 + \frac{- 7 x}{3}$

Step 1.1.1.2.3.1.2

Move the negative in front of the fraction.

$y \leq - 8 - \frac{7 x}{3}$

Step 1.1.2

Rearrange terms.

$y \leq - \frac{7 x}{3} - 8$

Step 1.1.3

Reorder terms.

$y \leq - (\frac{7}{3} x) - 8$

Step 1.1.4

Remove parentheses.

$y \leq - \frac{7}{3} x - 8$

Step 1.2

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

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Step 1.2.1

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

$m = - \frac{7}{3}$

$b = - 8$

Step 1.2.2

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

Slope: $- \frac{7}{3}$

y-intercept: $(0, - 8)$

Slope: $- \frac{7}{3}$

y-intercept: $(0, - 8)$

Step 1.3

Graph a solid line, then shade the area below the boundary line since $y$ is less than $- \frac{7}{3} x - 8$ .

$y \leq - \frac{7}{3} x - 8$

Step 2

Graph $x + 3 y \leq - 6$ .

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Step 2.1

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

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Step 2.1.1

Solve for $y$ .

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Step 2.1.1.1

Subtract $x$ from both sides of the inequality.

$3 y \leq - 6 - x$

Step 2.1.1.2

Divide each term in $3 y \leq - 6 - x$ by $3$ and simplify.

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Step 2.1.1.2.1

Divide each term in $3 y \leq - 6 - x$ by $3$ .

$\frac{3 y}{3} \leq \frac{- 6}{3} + \frac{- x}{3}$

Step 2.1.1.2.2

Simplify the left side.

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Step 2.1.1.2.2.1

Cancel the common factor of $3$ .

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Step 2.1.1.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} \leq \frac{- 6}{3} + \frac{- x}{3}$

Step 2.1.1.2.2.1.2

Divide $y$ by $1$ .

$y \leq \frac{- 6}{3} + \frac{- x}{3}$

Step 2.1.1.2.3

Simplify the right side.

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Step 2.1.1.2.3.1

Simplify each term.

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Step 2.1.1.2.3.1.1

Divide $- 6$ by $3$ .

$y \leq - 2 + \frac{- x}{3}$

Step 2.1.1.2.3.1.2

Move the negative in front of the fraction.

$y \leq - 2 - \frac{x}{3}$

Step 2.1.2

Rearrange terms.

$y \leq - \frac{x}{3} - 2$

Step 2.1.3

Reorder terms.

$y \leq - (\frac{1}{3} x) - 2$

Step 2.1.4

Remove parentheses.

$y \leq - \frac{1}{3} x - 2$

Step 2.2

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

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Step 2.2.1

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

$m = - \frac{1}{3}$

$b = - 2$

Step 2.2.2

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

Slope: $- \frac{1}{3}$

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

Slope: $- \frac{1}{3}$

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

Step 2.3

Graph a solid line, then shade the area below the boundary line since $y$ is less than $- \frac{1}{3} x - 2$ .

$y \leq - \frac{1}{3} x - 2$

Step 3

Plot each graph on the same coordinate system.

$7 x + 3 y \leq - 24$

$x + 3 y \leq - 6$

Step 4