Use your knowledge of x- and y-intercepts to choose the correct graph of the equation 3x+6y=6.
The Correct Answer and Explanation is :
To find the correct graph of the equation (3x + 6y = 6), we’ll start by determining the x- and y-intercepts. Intercepts are key points where the graph crosses the axes: the x-intercept is where the graph crosses the x-axis (y = 0), and the y-intercept is where the graph crosses the y-axis (x = 0).
Step 1: Find the X-Intercept
Set (y = 0) in the equation and solve for (x):
[
3x + 6(0) = 6 \Rightarrow 3x = 6 \Rightarrow x = 2
]
So, the x-intercept is at ((2, 0)).
Step 2: Find the Y-Intercept
Set (x = 0) in the equation and solve for (y):
[
3(0) + 6y = 6 \Rightarrow 6y = 6 \Rightarrow y = 1
]
So, the y-intercept is at ((0, 1)).
Step 3: Rewrite the Equation in Slope-Intercept Form
We can also rewrite the equation in slope-intercept form, (y = mx + b), to better understand its graph. Start by isolating (y):
[
3x + 6y = 6 \Rightarrow 6y = -3x + 6 \Rightarrow y = -\frac{1}{2}x + 1
]
This form tells us that the slope (m) is (-\frac{1}{2}), and the y-intercept (b) is 1. This means that the line has a downward slope, indicating it falls by (\frac{1}{2}) unit for each 1 unit increase in (x).
Step 4: Plot the Points and Graph the Line
To graph this line, plot the intercepts ((2, 0)) and ((0, 1)), then draw a line through these points. The slope of (-\frac{1}{2}) confirms that the line decreases as (x) increases.
Summary
The correct graph of (3x + 6y = 6) is a line passing through ((2, 0)) and ((0, 1)) with a negative slope. These intercepts and the slope match a line with an x-intercept at 2 and a y-intercept at 1, sloping downward from left to right.