Find the intercepts and graph the equation by plotting points. \( 2 x+3 y=6 \)

To find the intercepts and graph the equation \( 2x + 3y = 6 \), follow these steps: ### 1. Find the Intercepts **a. X-Intercept:** The x-intercept occurs where the graph crosses the x-axis. At this point, \( y = 0 \). \[ 2x + 3(0) = 6 \\ 2x = 6 \\ x = 3 \] **X-Intercept:** \( (3, 0) \) --- **b. Y-Intercept:** The y-intercept occurs where the graph crosses the y-axis. At this point, \( x = 0 \). \[ 2(0) + 3y = 6 \\ 3y = 6 \\ y = 2 \] **Y-Intercept:** \( (0, 2) \) ### 2. Plot the Intercepts on a Coordinate Plane 1. **Plot the X-Intercept:** Place a point at \( (3, 0) \). 2. **Plot the Y-Intercept:** Place a point at \( (0, 2) \). ### 3. Find Additional Points (Optional) To ensure accuracy, you can find additional points by choosing values for \( x \) or \( y \) and solving for the other variable. **Example 1:** Let \( x = 1 \) \[ 2(1) + 3y = 6 \\ 2 + 3y = 6 \\ 3y = 4 \\ y = \frac{4}{3} \approx 1.33 \] **Point:** \( (1, \frac{4}{3}) \) **Example 2:** Let \( y = 1 \) \[ 2x + 3(1) = 6 \\ 2x + 3 = 6 \\ 2x = 3 \\ x = \frac{3}{2} = 1.5 \] **Point:** \( \left( \frac{3}{2}, 1 \right) \) ### 4. Draw the Graph 1. **Draw the Axes:** Draw a horizontal x-axis and a vertical y-axis intersecting at the origin \( (0, 0) \). 2. **Plot the Points:** - \( (3, 0) \) — X-Intercept - \( (0, 2) \) — Y-Intercept - \( (1, \frac{4}{3}) \) — Additional Point - \( \left( \frac{3}{2}, 1 \right) \) — Additional Point 3. **Connect the Points:** Draw a straight line that passes through all the plotted points. This line represents the equation \( 2x + 3y = 6 \). ### 5. Visual Representation Below is a simple graphical representation: ``` y | | * | (0,2) | | * | (1.5,1) | | * | (3,0) | +------------------ x ``` - The points \( (3, 0) \) and \( (0, 2) \) are the intercepts. - The additional points help in drawing the accurate line. ### 6. Slope-Intercept Form (Optional) For better understanding, you can rewrite the equation in slope-intercept form (\( y = mx + b \)): \[ 2x + 3y = 6 \\ 3y = -2x + 6 \\ y = -\frac{2}{3}x + 2 \] - **Slope (\( m \))**: \( -\frac{2}{3} \) - **Y-Intercept (\( b \))**: \( 2 \) This means the line decreases by \( \frac{2}{3} \) units vertically for every 1 unit it moves horizontally to the right. ### Summary - **X-Intercept:** \( (3, 0) \) - **Y-Intercept:** \( (0, 2) \) - **Slope:** \( -\frac{2}{3} \) By plotting these points and using the slope, you can accurately graph the equation \( 2x + 3y = 6 \).