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Chapter 3: Problem 46

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

Short Answer

Expert verified
Plot the intercepts (4, 0) and (0, 2) and draw a line through them.

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. Substituting y = 0 into the equation gives:\( x + 2(0) = 4 \)This simplifies to:\( x = 4 \)So, the x-intercept is (4, 0).
02

Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. Substituting x = 0 into the equation gives:\( 0 + 2y = 4 \)This simplifies to:\( 2y = 4 \)Dividing both sides by 2, we get:\( y = 2 \)So, the y-intercept is (0, 2).
03

Plot the intercepts

Plot the points (4, 0) and (0, 2) on the coordinate plane. These are the x and y intercepts respectively.
04

Draw the line

Draw a straight line through the points (4, 0) and (0, 2). This line represents the equation \( x + 2y = 4 \).

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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** is the point where a line crosses the x-axis. To find it, we set y to 0 in the given equation. For example, given the equation \(x + 2y = 4\), we substitute y = 0:
  • \(x + 2(0) = 4\)
  • This simplifies to \(x = 4\)
So, the x-intercept is (4, 0). This point is crucial, as it tells us where our line touches the x-axis. Knowing how to find the x-intercept helps us graph linear equations effectively.
y-intercept
The **y-intercept** is the point where a line crosses the y-axis. To determine this, we set x to 0 in the equation. Using the same equation, \(x + 2y = 4\), substitute x = 0:
  • \(0 + 2y = 4\)
  • This simplifies to \(2y = 4\)
  • Dividing both sides by 2, we get \(y = 2\)
Thus the y-intercept is (0, 2). The y-intercept is essential for graphing as it shows where our line touches the y-axis. Mastering this concept allows us to construct more precise graphs.
coordinate plane
The **coordinate plane** is a two-dimensional surface divided by a horizontal line (x-axis) and a vertical line (y-axis). This plane is used to plot points, lines, and curves. Each point on the plane is described by an ordered pair (x, y). For instance, the points (4, 0) and (0, 2) are plotted as follows:
  • (4, 0): Move right 4 units along the x-axis and stay on the axis (because y = 0).
  • (0, 2): Move up 2 units along the y-axis and stay on the y-axis (because x = 0).
By understanding the coordinate plane, we can visually interpret relationships between variables.
linear equation
A **linear equation** is an equation that forms a straight line when graphed on the coordinate plane. The general form is \(Ax + By = C\). In our example, the equation \( x + 2y = 4\) is linear. Here's how to graph it:
  • First, find the x-intercept (4, 0) by setting y to 0.
  • Next, find the y-intercept (0, 2) by setting x to 0.
  • Plot these points on the coordinate plane.
  • Draw a straight line through both points.
This line represents all solutions to the linear equation. Learning to graph linear equations helps in understanding the relationship between variables visually.

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