Algebra Examples

$x + 8 y \leq 50$ $x \leq 30$ $y \geq 2$

Step 1

Graph $x + 8 y \leq 50$ .

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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 $x$ from both sides of the inequality.

$8 y \leq 50 - x$

Step 1.1.1.2

Divide each term in $8 y \leq 50 - x$ by $8$ and simplify.

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Step 1.1.1.2.1

Divide each term in $8 y \leq 50 - x$ by $8$ .

$\frac{8 y}{8} \leq \frac{50}{8} + \frac{- x}{8}$

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 $8$ .

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Step 1.1.1.2.2.1.1

Cancel the common factor.

$\frac{8 y}{8} \leq \frac{50}{8} + \frac{- x}{8}$

Step 1.1.1.2.2.1.2

Divide $y$ by $1$ .

$y \leq \frac{50}{8} + \frac{- x}{8}$

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

Cancel the common factor of $50$ and $8$ .

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Step 1.1.1.2.3.1.1.1

Factor $2$ out of $50$ .

$y \leq \frac{2 (25)}{8} + \frac{- x}{8}$

Step 1.1.1.2.3.1.1.2

Cancel the common factors.

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Step 1.1.1.2.3.1.1.2.1

Factor $2$ out of $8$ .

$y \leq \frac{2 \cdot 25}{2 \cdot 4} + \frac{- x}{8}$

Step 1.1.1.2.3.1.1.2.2

Cancel the common factor.

$y \leq \frac{2 \cdot 25}{2 \cdot 4} + \frac{- x}{8}$

Step 1.1.1.2.3.1.1.2.3

Rewrite the expression.

$y \leq \frac{25}{4} + \frac{- x}{8}$

Step 1.1.1.2.3.1.2

Move the negative in front of the fraction.

$y \leq \frac{25}{4} - \frac{x}{8}$

Step 1.1.2

Rearrange terms.

$y \leq - \frac{x}{8} + \frac{25}{4}$

Step 1.1.3

Reorder terms.

$y \leq - (\frac{1}{8} x) + \frac{25}{4}$

Step 1.1.4

Remove parentheses.

$y \leq - \frac{1}{8} x + \frac{25}{4}$

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 $y = m x + b$ .

$m = - \frac{1}{8}$

$b = \frac{25}{4}$

Step 1.2.2

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

Slope: $- \frac{1}{8}$

y-intercept: $(0, \frac{25}{4})$

Slope: $- \frac{1}{8}$

y-intercept: $(0, \frac{25}{4})$

Step 1.3

Graph a solid line, then shade the area below the boundary line since $y$ is less than $- \frac{1}{8} x + \frac{25}{4}$ .

$y \leq - \frac{1}{8} x + \frac{25}{4}$

Step 2

Graph $x \leq 30$ .

$x \leq 30$

Step 3

Graph $y \geq 2$ .

$y \geq 2$

Step 4

Plot each graph on the same coordinate system.

$x + 8 y \leq 50$

$x \leq 30$

$y \geq 2$

Step 5