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Chapter 11: Problem 171

In the following exercises, graph using the intercepts. $$ 2 x+4 y=12 $$

Short Answer

Expert verified
The x-intercept is (6, 0) and the y-intercept is (0, 3). Plot these points and draw a line through them to graph the equation.

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x.Starting with the equation:\[2x + 4y = 12\]Set \(y = 0\):\[2x + 4(0) = 12\]Simplify to get:\[2x = 12\]Divide both sides by 2:\[x = 6\]So, the x-intercept is (6, 0).
02

Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y.Starting with the equation:\[2x + 4y = 12\]Set \(x = 0\):\[2(0) + 4y = 12\]Simplify to get:\[4y = 12\]Divide both sides by 4:\[y = 3\]So, the y-intercept is (0, 3).
03

Plot the intercepts

Plot the points (6, 0) and (0, 3) on the coordinate plane. These points are the intercepts of the equation.
04

Draw the graph

Draw a straight line passing through the points (6, 0) and (0, 3). This line represents the graph of the equation \(2x + 4y = 12\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

x-intercept
The x-intercept of a graph is where the line crosses the x-axis. To find the x-intercept, you simply set the value of y to 0 in the equation and solve for x. For example, in the equation \ 2x + 4y = 12, we start by setting y to 0: \[2x + 4(0) = 12\] This simplifies to \ 2x = 12. Next, we divide both sides by 2: \[x = 6\] So, the x-intercept is at the point (6, 0). This intersection tells us that when y is 0, x must be 6 on the graph.
y-intercept
The y-intercept of a graph is where the line crosses the y-axis. To find the y-intercept, you set the value of x to 0 in the equation and solve for y. In our equation \ 2x + 4y = 12, set x to 0: \[2(0) + 4y = 12\] This simplifies to \ 4y = 12. Now, divide both sides by 4: \[y = 3\] So, the y-intercept is at the point (0, 3). This tells us that when x is 0, y equals 3 on the graph.
coordinate plane
The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It's defined by two perpendicular number lines: the x-axis (horizontal) and y-axis (vertical). The point where these axes intersect is called the origin, represented by (0, 0). On this plane, every point is identified by a pair of numbers known as coordinates. The first number is the x-coordinate and the second is the y-coordinate, typically written as (x, y). For example, the point (6, 0) sits on the x-axis, six units to the right of the origin. Similarly, the point (0, 3) sits on the y-axis, three units above the origin.
plotting points
Plotting points on the coordinate plane is essential for graphing equations. To plot a point, you need its coordinates (x, y). Start at the origin (0, 0). Move horizontally to the x-coordinate and then vertically to the y-coordinate. For instance, to plot (6, 0), start at the origin and move 6 units to the right. For (0, 3), start again at the origin, but move 3 units up. Once both points are plotted, you can draw a line through them to represent the graph of the equation. This line shows every point (x, y) that satisfies the equation 2x + 4y = 12.

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Most popular questions from this chapter

Complete the table to find three solutions to the equation \(4 x+y=8\)

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