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Chapter 11: Problem 175

In the following exercises, graph using the intercepts. $$ 2 x-5 y=-20 $$

Short Answer

Expert verified
The x-intercept is (-10, 0) and the y-intercept is (0, 4).

Step by step solution

01

Find the x-intercept

The x-intercept occurs when y = 0. Substitute y = 0 into the equation and solve for x:\[ 2x - 5(0) = -20 \]\[ 2x = -20 \]\[ x = -10 \]So, the x-intercept is (-10, 0).
02

Find the y-intercept

The y-intercept occurs when x = 0. Substitute x = 0 into the equation and solve for y:\[ 2(0) - 5y = -20 \]\[ -5y = -20 \]\[ y = 4 \]So, the y-intercept is (0, 4).
03

Plot both intercepts on the graph

Plot the points (-10, 0) and (0, 4) on the coordinate plane. These points represent the x-intercept and y-intercept respectively.
04

Draw the line

Draw a straight line through the points (-10, 0) and (0, 4) to represent the equation 2x - 5y = -20 on the graph.

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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 an essential concept when graphing linear equations. It is the point where the line crosses the x-axis. To find this point, set y = 0 and solve for x. In the equation \(2x - 5y = -20\), substituting y = 0 gives us:

\[ 2x - 5(0) = -20 \]
\[ 2x = -20 \]
\[ x = -10 \]

Thus, the x-intercept is the point \(-10, 0\). This means the line crosses the x-axis at -10. Remember, finding the intercepts helps to identify key points through which the line will pass on the graph.
y-intercept
The y-intercept is another crucial concept for graphing linear equations. It represents the point where the line crosses the y-axis. To determine the y-intercept, set x = 0 and solve for y. For the equation \(2x - 5y = -20\), substitute x = 0:

\[ 2(0) - 5y = -20 \]
\[ -5y = -20 \]
\[ y = 4 \]

So, the y-intercept is the point \(0, 4\). This indicates that the line crosses the y-axis at 4. Finding the intercepts is crucial for accurately drawing the line on a coordinate plane.
coordinate plane
A coordinate plane is a two-dimensional surface formed by two perpendicular lines called axes. These axes are the x-axis (horizontal) and the y-axis (vertical). The point where they intersect is called the origin, which is \((0, 0)\). Each point on the coordinate plane is represented by an ordered pair \((x, y)\). The x-coordinate shows the position along the x-axis, and the y-coordinate shows the position along the y-axis.

To graph a line using intercepts:
  • Identify the x-intercept and y-intercept.
  • Plot these points on the coordinate plane.
  • Draw a line passing through both intercepts.
Using a coordinate plane makes it easier to visualize how equations behave and interact with each other.
plotting points
Plotting points is a fundamental skill for graphing equations. To plot a point, find its position using its coordinates \((x, y)\).
  • For the x-coordinate, move horizontally from the origin (positive moves right, negative moves left).
  • For the y-coordinate, move vertically from the origin (positive moves up, negative moves down).
  • Mark the point where these two movements intersect.
In the given exercise:
  • Plot the x-intercept \((-10, 0)\) by moving 10 units left and placing the point on the x-axis.
  • Plot the y-intercept \((0, 4)\) by moving 4 units up and placing the point on the y-axis.
Once both points are plotted, draw a straight line passing through them to visualize the graph of the equation \(2x - 5y = -20\). Accurate plotting ensures a precise representation of the linear equation.

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