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Use the intercepts to graph the equation. \( x+3 y=6 \)

Ask by Maxwell Hall. in the United States
Jan 21,2025

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To graph the equation \( x + 3y = 6 \), find the x-intercept at \( (6, 0) \) and the y-intercept at \( (0, 2) \). Plot these points and draw a straight line through them.

Solución

To graph the equation \( x + 3y = 6 \) using its intercepts, follow these steps: ### 1. Find the **x-intercept** The **x-intercept** occurs where the graph crosses the x-axis. At this point, the value of \( y \) is **0**. **Steps:** - Set \( y = 0 \) in the equation: \[ x + 3(0) = 6 \implies x = 6 \] - **x-intercept:** \( (6, 0) \) ### 2. Find the **y-intercept** The **y-intercept** occurs where the graph crosses the y-axis. At this point, the value of \( x \) is **0**. **Steps:** - Set \( x = 0 \) in the equation: \[ 0 + 3y = 6 \implies y = 2 \] - **y-intercept:** \( (0, 2) \) ### 3. Plot the Intercepts - Plot the **x-intercept** at \( (6, 0) \). - Plot the **y-intercept** at \( (0, 2) \). ### 4. Draw the Graph - Draw a straight line passing through the two plotted points \( (6, 0) \) and \( (0, 2) \). - This line represents the equation \( x + 3y = 6 \). ### 5. Graphical Representation Below is a simple graphical representation of the line: ``` y | 2 | * | | | | 0 |------------------------ x 0 6 ``` - The point \( (0, 2) \) is the **y-intercept** marked with an asterisk (*). - The point \( (6, 0) \) is the **x-intercept**. - The straight line connects these two intercepts. ### 6. Slope-Intercept Form (Optional) For additional insight, you can rewrite the equation in **slope-intercept form** \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. **Steps:** \[ x + 3y = 6 \\ 3y = -x + 6 \\ y = -\frac{1}{3}x + 2 \] - **Slope (\( m \))**: \( -\frac{1}{3} \) - **y-intercept (\( b \))**: \( 2 \) This form confirms that the line crosses the y-axis at \( (0, 2) \) and has a slope of \( -\frac{1}{3} \), meaning it decreases by 1 unit in \( y \) for every 3 units increase in \( x \). ### Summary - **x-intercept:** \( (6, 0) \) - **y-intercept:** \( (0, 2) \) - **Slope-intercept form:** \( y = -\frac{1}{3}x + 2 \) - **Graph:** A straight line passing through \( (6, 0) \) and \( (0, 2) \) By following these steps, you can accurately graph the equation \( x + 3y = 6 \) using its intercepts.

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Bonus Knowledge

To graph the equation \( x + 3y = 6 \), start by finding the intercepts. For the x-intercept, set \( y = 0 \): this gives \( x = 6 \), so the point is \( (6, 0) \). For the y-intercept, set \( x = 0 \): that results in \( 3y = 6 \), leading to \( y = 2 \), giving the point \( (0, 2) \). Plot these two points on the graph, and then draw a line through them to complete your representation! To make your graph neat, you can also calculate additional points by substituting other values of \( x \) or \( y \). For example, if you substitute \( x = 3 \), you find \( y = 1 \), yielding the point \( (3, 1) \). Adding more points ensures that your line is accurate and visually appealing! Don’t forget to label your axes and indicate the scale for clarity. Happy graphing!

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