Algebra Examples

$3 x + y \leq 1$ $x - y \leq 3$

Step 1

Graph $3 x + y \leq 1$ .

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Step 1.1

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

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Step 1.1.1

Subtract $3 x$ from both sides of the inequality.

$y \leq 1 - 3 x$

Step 1.1.2

Rearrange terms.

$y \leq - 3 x + 1$

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 $y = m x + b$ .

$m = - 3$

$b = 1$

Step 1.2.3

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

Slope: $- 3$

y-intercept: $(0, 1)$

Slope: $- 3$

y-intercept: $(0, 1)$

Step 1.3

Graph a solid line, then shade the area below the boundary line since $y$ is less than $- 3 x + 1$ .

$y \leq - 3 x + 1$

Step 2

Graph $x - y \leq 3$ .

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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.

$- y \leq 3 - x$

Step 2.1.1.2

Divide each term in $- y \leq 3 - x$ by $- 1$ and simplify.

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Step 2.1.1.2.1

Divide each term in $- y \leq 3 - x$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} \geq \frac{3}{- 1} + \frac{- x}{- 1}$

Step 2.1.1.2.2

Simplify the left side.

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Step 2.1.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} \geq \frac{3}{- 1} + \frac{- x}{- 1}$

Step 2.1.1.2.2.2

Divide $y$ by $1$ .

$y \geq \frac{3}{- 1} + \frac{- x}{- 1}$

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 $3$ by $- 1$ .

$y \geq - 3 + \frac{- x}{- 1}$

Step 2.1.1.2.3.1.2

Dividing two negative values results in a positive value.

$y \geq - 3 + \frac{x}{1}$

Step 2.1.1.2.3.1.3

Divide $x$ by $1$ .

$y \geq - 3 + x$

Step 2.1.2

Rearrange terms.

$y \geq x - 3$

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 $y = m x + b$ .

$m = 1$

$b = - 3$

Step 2.2.3

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

Slope: $1$

y-intercept: $(0, - 3)$

Slope: $1$

y-intercept: $(0, - 3)$

Step 2.3

Graph a solid line, then shade the area above the boundary line since $y$ is greater than $x - 3$ .

$y \geq x - 3$

Step 3

Plot each graph on the same coordinate system.

$3 x + y \leq 1$

$x - y \leq 3$

Step 4