Graph 2x+3y<=12 3x-5y>=15

$2 x + 3 y \leq 12$ $3 x - 5 y \geq 15$

Step 1

Graph $2 x + 3 y \leq 12$ .

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

Solve for $y$ .

Subtract $2 x$ from both sides of the inequality.

$3 y \leq 12 - 2 x$

Divide each term in $3 y \leq 12 - 2 x$ by $3$ and simplify.

Divide each term in $3 y \leq 12 - 2 x$ by $3$ .

$\frac{3 y}{3} \leq \frac{12}{3} + \frac{- 2 x}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y}{3} \leq \frac{12}{3} + \frac{- 2 x}{3}$

Divide $y$ by $1$ .

$y \leq \frac{12}{3} + \frac{- 2 x}{3}$

Simplify the right side.

Simplify each term.

Divide $12$ by $3$ .

$y \leq 4 + \frac{- 2 x}{3}$

Move the negative in front of the fraction.

$y \leq 4 - \frac{2 x}{3}$

Rearrange terms.

$y \leq - \frac{2 x}{3} + 4$

Reorder terms.

$y \leq - (\frac{2}{3} x) + 4$

Remove parentheses.

$y \leq - \frac{2}{3} x + 4$

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

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

$m = - \frac{2}{3}$

$b = 4$

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

Slope: $- \frac{2}{3}$

y-intercept: $(0, 4)$

Slope: $- \frac{2}{3}$

y-intercept: $(0, 4)$

Graph a solid line, then shade the area below the boundary line since $y$ is less than $- \frac{2}{3} x + 4$ .

$y \leq - \frac{2}{3} x + 4$

Step 2

Graph $3 x - 5 y \geq 15$ .

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

Solve for $y$ .

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

$- 5 y \geq 15 - 3 x$

Divide each term in $- 5 y \geq 15 - 3 x$ by $- 5$ and simplify.

Divide each term in $- 5 y \geq 15 - 3 x$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 y}{- 5} \leq \frac{15}{- 5} + \frac{- 3 x}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 y}{- 5} \leq \frac{15}{- 5} + \frac{- 3 x}{- 5}$

Divide $y$ by $1$ .

$y \leq \frac{15}{- 5} + \frac{- 3 x}{- 5}$

Simplify the right side.

Simplify each term.

Divide $15$ by $- 5$ .

$y \leq - 3 + \frac{- 3 x}{- 5}$

Dividing two negative values results in a positive value.

$y \leq - 3 + \frac{3 x}{5}$

Rearrange terms.

$y \leq \frac{3 x}{5} - 3$

Reorder terms.

$y \leq \frac{3}{5} x - 3$

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

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

$m = \frac{3}{5}$

$b = - 3$

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

Slope: $\frac{3}{5}$

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

Slope: $\frac{3}{5}$

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

Graph a solid line, then shade the area below the boundary line since $y$ is less than $\frac{3}{5} x - 3$ .

$y \leq \frac{3}{5} x - 3$

Step 3

Plot each graph on the same coordinate system.

$2 x + 3 y \leq 12$

$3 x - 5 y \geq 15$

Step 4

Graph 2x+3y<=12 3x-5y>=15

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