Graph 3x+4y=6

$3 x + 4 y = 6$

Step 1

Solve for $y$ .

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

$4 y = 6 - 3 x$

Divide each term in $4 y = 6 - 3 x$ by $4$ and simplify.

Divide each term in $4 y = 6 - 3 x$ by $4$ .

$\frac{4 y}{4} = \frac{6}{4} + \frac{- 3 x}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{6}{4} + \frac{- 3 x}{4}$

Divide $y$ by $1$ .

$y = \frac{6}{4} + \frac{- 3 x}{4}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $6$ and $4$ .

Factor $2$ out of $6$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 2

Rewrite in slope-intercept form.

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Reorder $\frac{3}{2}$ and $- \frac{3 x}{4}$ .

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

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

Reorder terms.

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

Remove parentheses.

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

Step 3

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

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

$m = - \frac{3}{4}$

$b = \frac{3}{2}$

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

Slope: $- \frac{3}{4}$

y-intercept: $(0, \frac{3}{2})$

Slope: $- \frac{3}{4}$

y-intercept: $(0, \frac{3}{2})$

Step 4

Any line can be graphed using two points. Select two $x$ values, and plug them into the equation to find the corresponding $y$ values.

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

Reorder $\frac{3}{2}$ and $- \frac{3 x}{4}$ .

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

Reorder terms.

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

Remove parentheses.

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

Find the x-intercept.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - \frac{3}{4} x + \frac{3}{2}$

Solve the equation.

Rewrite the equation as $- \frac{3}{4} x + \frac{3}{2} = 0$ .

$- \frac{3}{4} x + \frac{3}{2} = 0$

Simplify each term.

Combine $x$ and $\frac{3}{4}$ .

$- \frac{x \cdot 3}{4} + \frac{3}{2} = 0$

Move $3$ to the left of $x$ .

$- \frac{3 x}{4} + \frac{3}{2} = 0$

Subtract $\frac{3}{2}$ from both sides of the equation.

$- \frac{3 x}{4} = - \frac{3}{2}$

Multiply both sides of the equation by $- \frac{4}{3}$ .

$- \frac{4}{3} (- \frac{3 x}{4}) = - \frac{4}{3} (- \frac{3}{2})$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{4}{3} (- \frac{3 x}{4})$ .

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$\frac{- 4}{3} (- \frac{3 x}{4}) = - \frac{4}{3} (- \frac{3}{2})$

Move the leading negative in $- \frac{3 x}{4}$ into the numerator.

$\frac{- 4}{3} \cdot \frac{- 3 x}{4} = - \frac{4}{3} (- \frac{3}{2})$

Factor $4$ out of $- 4$ .

$\frac{4 (- 1)}{3} \cdot \frac{- 3 x}{4} = - \frac{4}{3} (- \frac{3}{2})$

Cancel the common factor.

$\frac{4 \cdot - 1}{3} \cdot \frac{- 3 x}{4} = - \frac{4}{3} (- \frac{3}{2})$

Rewrite the expression.

$\frac{- 1}{3} (- 3 x) = - \frac{4}{3} (- \frac{3}{2})$

Cancel the common factor of $3$ .

Factor $3$ out of $- 3 x$ .

$\frac{- 1}{3} (3 (- x)) = - \frac{4}{3} (- \frac{3}{2})$

Cancel the common factor.

$\frac{- 1}{3} (3 (- x)) = - \frac{4}{3} (- \frac{3}{2})$

Rewrite the expression.

$- - x = - \frac{4}{3} (- \frac{3}{2})$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{4}{3} (- \frac{3}{2})$

Multiply $x$ by $1$ .

$x = - \frac{4}{3} (- \frac{3}{2})$

Simplify the right side.

Simplify $- \frac{4}{3} (- \frac{3}{2})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$x = \frac{- 4}{3} (- \frac{3}{2})$

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$x = \frac{- 4}{3} \cdot \frac{- 3}{2}$

Factor $2$ out of $- 4$ .

$x = \frac{2 (- 2)}{3} \cdot \frac{- 3}{2}$

Cancel the common factor.

$x = \frac{2 \cdot - 2}{3} \cdot \frac{- 3}{2}$

Rewrite the expression.

$x = \frac{- 2}{3} \cdot - 3$

Cancel the common factor of $3$ .

Factor $3$ out of $- 3$ .

$x = \frac{- 2}{3} \cdot (3 (- 1))$

Cancel the common factor.

$x = \frac{- 2}{3} \cdot (3 \cdot - 1)$

Rewrite the expression.

$x = - 2 \cdot - 1$

Multiply $- 2$ by $- 1$ .

$x = 2$

x-intercept(s) in point form.

x-intercept(s): $(2, 0)$

Find the y-intercept.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - \frac{3}{4} \cdot 0 + \frac{3}{2}$

Solve the equation.

Remove parentheses.

$y = - \frac{3}{4} \cdot 0 + \frac{3}{2}$

Simplify $- \frac{3}{4} \cdot 0 + \frac{3}{2}$ .

Multiply $- \frac{3}{4} \cdot 0$ .

Multiply $0$ by $- 1$ .

$y = 0 (\frac{3}{4}) + \frac{3}{2}$

Multiply $0$ by $\frac{3}{4}$ .

$y = 0 + \frac{3}{2}$

Add $0$ and $\frac{3}{2}$ .

$y = \frac{3}{2}$

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{3}{2})$

Create a table of the $x$ and $y$ values.

$\begin{array}{c} x & y \\ 0 & \frac{3}{2} \\ 2 & 0 \end{array}$

Step 5

Graph the line using the slope and the y-intercept, or the points.

Slope: $- \frac{3}{4}$

y-intercept: $(0, \frac{3}{2})$

$\begin{array}{c} x & y \\ 0 & \frac{3}{2} \\ 2 & 0 \end{array}$

Step 6

Graph 3x+4y=6

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