Algebra Examples

$x + y \leq 30$ $2.5 x + 2.75 y \geq 44$

Step 1

Graph $x + y \leq 30$ .

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Step 1.1

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

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Step 1.1.1

Subtract $x$ from both sides of the inequality.

$y \leq 30 - x$

Step 1.1.2

Rearrange terms.

$y \leq - x + 30$

Step 1.2

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

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Step 1.2.1

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 1.2.2

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

$m = - 1$

$b = 30$

Step 1.2.3

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

Slope: $- 1$

y-intercept: $(0, 30)$

Slope: $- 1$

y-intercept: $(0, 30)$

Step 1.3

Graph a solid line, then shade the area below the boundary line since $y$ is less than $- x + 30$ .

$y \leq - x + 30$

Step 2

Graph $2.5 x + 2.75 y \geq 44$ .

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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 $2.5 x$ from both sides of the inequality.

$2.75 y \geq 44 - 2.5 x$

Step 2.1.1.2

Divide each term in $2.75 y \geq 44 - 2.5 x$ by $2.75$ and simplify.

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Step 2.1.1.2.1

Divide each term in $2.75 y \geq 44 - 2.5 x$ by $2.75$ .

$\frac{2.75 y}{2.75} \geq \frac{44}{2.75} + \frac{- 2.5 x}{2.75}$

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 $2.75$ .

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Step 2.1.1.2.2.1.1

Cancel the common factor.

$\frac{2.75 y}{2.75} \geq \frac{44}{2.75} + \frac{- 2.5 x}{2.75}$

Step 2.1.1.2.2.1.2

Divide $y$ by $1$ .

$y \geq \frac{44}{2.75} + \frac{- 2.5 x}{2.75}$

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 $44$ by $2.75$ .

$y \geq 16 + \frac{- 2.5 x}{2.75}$

Step 2.1.1.2.3.1.2

Move the negative in front of the fraction.

$y \geq 16 - \frac{2.5 x}{2.75}$

Step 2.1.1.2.3.1.3

Factor $2.5$ out of $2.5 x$ .

$y \geq 16 - \frac{2.5 (x)}{2.75}$

Step 2.1.1.2.3.1.4

Factor $2.75$ out of $2.75$ .

$y \geq 16 - \frac{2.5 (x)}{2.75 (1)}$

Step 2.1.1.2.3.1.5

Separate fractions.

$y \geq 16 - (\frac{2.5}{2.75} \cdot \frac{x}{1})$

Step 2.1.1.2.3.1.6

Divide $2.5$ by $2.75$ .

$y \geq 16 - (0.\overline{90} \frac{x}{1})$

Step 2.1.1.2.3.1.7

Divide $x$ by $1$ .

$y \geq 16 - (0.\overline{90} x)$

Step 2.1.1.2.3.1.8

Multiply $0.\overline{90}$ by $- 1$ .

$y \geq 16 - 0.\overline{90} x$

Step 2.1.2

Rearrange terms.

$y \geq - 0.\overline{90} x + 16$

Step 2.2

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

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Step 2.2.1

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 2.2.2

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

$m = - 0.\overline{90}$

$b = 16$

Step 2.2.3

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

Slope: $- 0.\overline{90}$

y-intercept: $(0, 16)$

Slope: $- 0.\overline{90}$

y-intercept: $(0, 16)$

Step 2.3

Graph a solid line, then shade the area above the boundary line since $y$ is greater than $- 0.\overline{90} x + 16$ .

$y \geq - 0.\overline{90} x + 16$

Step 3

Plot each graph on the same coordinate system.

$x + y \leq 30$

$2.5 x + 2.75 y \geq 44$

Step 4