GRAPHING AN EQUATION Graph the linear equation. \( \begin{array}{lll}\text { 12. }-18 x+9 y=72 & \text { 13. } 16 x-4 y=2 & \text { 14. } \frac{1}{4} x+\frac{3}{4} y=1\end{array} \)

To graph each of the given linear equations, we'll follow these steps: 1. **Rewrite the equation in slope-intercept form** (\( y = mx + b \)), where \( m \) is the slope and \( b \) is the y-intercept. 2. **Identify the slope and y-intercept** from the equation. 3. **Determine the x-intercept** by setting \( y = 0 \) and solving for \( x \). 4. **Plot the intercepts** on a graph and draw the line through these points. Let's apply these steps to each equation: --- ### **12. \( -18x + 9y = 72 \)** **Step 1: Simplify the Equation** Divide every term by 9 to make the numbers smaller: \[ -18x + 9y = 72 \\ \Rightarrow \frac{-18x}{9} + \frac{9y}{9} = \frac{72}{9} \\ \Rightarrow -2x + y = 8 \] **Step 2: Solve for \( y \) (Slope-Intercept Form)** \[ y = 2x + 8 \] **Step 3: Identify Slope and Y-Intercept** - **Slope (\( m \))**: 2 - **Y-Intercept (\( b \))**: 8 (Point \( (0, 8) \)) **Step 4: Find X-Intercept** Set \( y = 0 \): \[ 0 = 2x + 8 \\ 2x = -8 \\ x = -4 \] - **X-Intercept**: \( (-4, 0) \) **Step 5: Plot and Draw the Line** - Plot the points \( (0, 8) \) and \( (-4, 0) \). - Draw a straight line through these points. --- ### **13. \( 16x - 4y = 2 \)** **Step 1: Simplify the Equation** Divide every term by 2: \[ 16x - 4y = 2 \\ \Rightarrow \frac{16x}{2} - \frac{4y}{2} = \frac{2}{2} \\ \Rightarrow 8x - 2y = 1 \] **Step 2: Solve for \( y \) (Slope-Intercept Form)** \[ 8x - 2y = 1 \\ -2y = -8x + 1 \\ y = 4x - \frac{1}{2} \] **Step 3: Identify Slope and Y-Intercept** - **Slope (\( m \))**: 4 - **Y-Intercept (\( b \))**: \( -\frac{1}{2} \) (Point \( (0, -0.5) \)) **Step 4: Find X-Intercept** Set \( y = 0 \): \[ 0 = 4x - \frac{1}{2} \\ 4x = \frac{1}{2} \\ x = \frac{1}{8} \] - **X-Intercept**: \( \left( \frac{1}{8}, 0 \right) \) **Step 5: Plot and Draw the Line** - Plot the points \( \left(0, -\frac{1}{2}\right) \) and \( \left(\frac{1}{8}, 0\right) \). - Draw a straight line through these points. --- ### **14. \( \frac{1}{4}x + \frac{3}{4}y = 1 \)** **Step 1: Eliminate Fractions** Multiply every term by 4 to simplify: \[ \frac{1}{4}x + \frac{3}{4}y = 1 \\ \Rightarrow 4 \times \frac{1}{4}x + 4 \times \frac{3}{4}y = 4 \times 1 \\ \Rightarrow x + 3y = 4 \] **Step 2: Solve for \( y \) (Slope-Intercept Form)** \[ x + 3y = 4 \\ 3y = -x + 4 \\ y = -\frac{1}{3}x + \frac{4}{3} \] **Step 3: Identify Slope and Y-Intercept** - **Slope (\( m \))**: \( -\frac{1}{3} \) - **Y-Intercept (\( b \))**: \( \frac{4}{3} \) (Point \( \left(0, \frac{4}{3}\right) \)) **Step 4: Find X-Intercept** Set \( y = 0 \): \[ 0 = -\frac{1}{3}x + \frac{4}{3} \\ \frac{1}{3}x = \frac{4}{3} \\ x = 4 \] - **X-Intercept**: \( (4, 0) \) **Step 5: Plot and Draw the Line** - Plot the points \( \left(0, \frac{4}{3}\right) \) and \( (4, 0) \). - Draw a straight line through these points. --- ### **Graphing Summary** For each equation: 1. **Plot the Y-Intercept**: Start by plotting the y-intercept on the y-axis. 2. **Plot the X-Intercept**: Next, plot the x-intercept on the x-axis. 3. **Draw the Line**: Use a ruler to draw a straight line that passes through both intercepts. 4. **Determine Additional Points (Optional)**: To ensure accuracy, you can plot additional points by choosing values for \( x \) or \( y \) and solving for the other variable. By following these steps, you can accurately graph each of the linear equations provided.