\( \mid<\quad \) Find the intercepts of \( 6 y-42=-21 x \) and then graph the line.

To find the intercepts of the equation \(6y - 42 = -21x\) and graph the line, follow these steps: ## 1. Find the **x-intercept** The **x-intercept** is the point where the line crosses the x-axis. At this point, the value of \(y\) is **0**. **Steps:** 1. Set \(y = 0\) in the equation. 2. Solve for \(x\). **Calculation:** \[ 6(0) - 42 = -21x \\ -42 = -21x \\ x = \frac{-42}{-21} = 2 \] **x-intercept:** \((2,\ 0)\) ## 2. Find the **y-intercept** The **y-intercept** is the point where the line crosses the y-axis. At this point, the value of \(x\) is **0**. **Steps:** 1. Set \(x = 0\) in the equation. 2. Solve for \(y\). **Calculation:** \[ 6y - 42 = -21(0) \\ 6y - 42 = 0 \\ 6y = 42 \\ y = 7 \] **y-intercept:** \((0,\ 7)\) ## 3. Graphing the Line To graph the line: 1. **Plot the intercepts:** - Plot the **x-intercept** at \((2,\ 0)\). - Plot the **y-intercept** at \((0,\ 7)\). 2. **Draw the line:** - Use a ruler to draw a straight line passing through both intercepts. - Extend the line in both directions and add arrows to indicate that it continues indefinitely. 3. **Label the intercepts:** - Label the points \((2,\ 0)\) and \((0,\ 7)\) for clarity. ### Visual Representation: While I can't provide a visual graph here, here's a description of how it would appear: - The line crosses the x-axis at \((2, 0)\) and the y-axis at \((0, 7)\). - Since the slope is negative, the line descends from left to right. Alternatively, you can use graphing tools like [Desmos](https://www.desmos.com/) or graph paper to plot the points and draw the line accurately. ## 4. Slope-Intercept Form (Optional) For additional insight, you can rewrite the equation in the **slope-intercept form** (\(y = mx + b\)), where \(m\) is the slope and \(b\) is the y-intercept. **Steps:** \[ 6y - 42 = -21x \\ 6y = -21x + 42 \\ y = \frac{-21}{6}x + 7 \\ y = -\frac{7}{2}x + 7 \] **Interpretation:** - **Slope (\(m\))**: \(-\frac{7}{2}\) - This means for every increase of 2 units in \(x\), \(y\) decreases by 7 units. - **y-intercept (\(b\))**: \(7\) - Confirms the y-intercept point \((0,\ 7)\) found earlier. ## Summary - **x-intercept:** \((2,\ 0)\) - **y-intercept:** \((0,\ 7)\) - **Slope:** \(-\frac{7}{2}\) - **Graph:** A straight line passing through \((2,\ 0)\) and \((0,\ 7)\), descending from left to right. Feel free to plot these points on graph paper or use an online graphing tool to visualize the line.