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Chapter 7: Problem 12

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ 5 x+2 y=10 $$

Short Answer

Expert verified
The x-intercept is (2, 0) and the y-intercept is (0, 5).

Step by step solution

01

- Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. Substitute x = 0 into the equation: 5(0) + 2y = 10 0 + 2y = 10 Solve for y: 2y = 10 y = 5. So, the y-intercept is (0, 5).
02

- Find the x-intercept

To find the x-intercept, set y to 0 and solve for x. Substitute y = 0 into the equation: 5x + 2(0) = 10 5x + 0 = 10 Solve for x: 5x = 10 x = 2. So, the x-intercept is (2, 0).
03

- Plot the intercepts on the graph

Plot the points (0, 5) and (2, 0) on the coordinate plane. These are the y-intercept and x-intercept, respectively.
04

- Draw the line

Draw a straight line passing through the points (0, 5) and (2, 0). This is the graph of the equation 5x + 2y = 10.

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Key Concepts

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

x-intercept
To find the x-intercept of a linear equation, we set the y-value to 0 and solve for x. This point is where the line crosses the x-axis.
For our example equation, 5x + 2y = 10, we can find the x-intercept by substituting y = 0:

5x + 2(0) = 10
5x = 10
So, x = 2.

The x-intercept is (2, 0). This means that when y = 0, x = 2, and the line crosses the x-axis at the point (2, 0).
Finding the x-intercept is valuable because it helps us understand where the line of the equation intersects the horizontal axis.
y-intercept
Finding the y-intercept involves setting the x-value to 0 and solving for y. This point is where the line crosses the y-axis.
For our example equation, 5x + 2y = 10, we substitute x = 0 to find the y-intercept:

5(0) + 2y = 10
0 + 2y = 10
So, 2y = 10
And y = 5.

The y-intercept is (0, 5). This means that when x = 0, y = 5, and the line crosses the y-axis at the point (0, 5).
Knowing the y-intercept helps us understand where the line intersects the vertical axis and can provide a starting point for graphing the equation.
graphing
Graphing a linear equation involves plotting points on a coordinate plane and drawing a line through those points. The intercepts we've found are crucial for this.
For the equation 5x + 2y = 10, we have the points (0, 5) and (2, 0) from the y-intercept and x-intercept, respectively:
  • Plot the point (0, 5) on the y-axis.
  • Next, plot the point (2, 0) on the x-axis.

Once both points are plotted, draw a straight line passing through them.
This line represents all the solutions to the equation 5x + 2y = 10.
Graphing helps visualize how changes in x and y affect the equation and how the line behaves in the coordinate plane.

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