6 to 9. Graph x−2y=2. Solve for the x-intercept and the y-intercept then graph the equation. x-intercept y-intercept
The Correct Answer and Explanation is:
To solve the equation x−2y=2x – 2y = 2, we first find the x-intercept and y-intercept, and then use them to graph the line.
1. Finding the x-intercept
Set y=0y = 0 in the equation:
x−2(0)=2⇒x=2x – 2(0) = 2 \Rightarrow x = 2
x-intercept: (2, 0)
2. Finding the y-intercept
Set x=0x = 0 in the equation:
0−2y=2⇒−2y=2⇒y=−10 – 2y = 2 \Rightarrow -2y = 2 \Rightarrow y = -1
y-intercept: (0, -1)
Explanation
To understand the behavior of a linear equation like x−2y=2x – 2y = 2, we often begin by identifying where the line intersects the axes. These intersections provide anchor points that help in graphing the line accurately.
The x-intercept is the point where the graph crosses the x-axis. At this location, the y-coordinate is always zero. By substituting y=0y = 0 into the equation and solving for xx, we determine that the line crosses the x-axis at (2, 0). Similarly, the y-intercept occurs where the line crosses the y-axis, and this happens when x=0x = 0. Solving for yy with x=0x = 0 gives us the point (0, -1).
With these two points, one can draw a straight line by connecting them on a coordinate plane. Any other point that lies on this line will also satisfy the equation x−2y=2x – 2y = 2. This technique is fundamental in algebra because it visualizes the solution set of a linear equation. Recognizing how coefficients affect the steepness and direction of a line is essential when analyzing or comparing equations. In this case, the line slopes upward from left to right, indicating a positive relationship between xx and yy.
