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Chapter 0: Problem 15

Determine the intercepts of the graphs of the following equations. $$ x-5 y=0 $$

Short Answer

Expert verified
The x-intercept and y-intercept are both at (0, 0).

Step by step solution

01

Understand the Equation

The given equation is in the standard linear form: \( x - 5y = 0 \). This can be used to find both the x-intercept and the y-intercept of the graph.
02

Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation and solve for \( x \): \[ x - 5(0) = 0 \]\[ x = 0 \]So, the x-intercept is at the point \( (0, 0) \).
03

Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation and solve for \( y \): \[ 0 - 5y = 0 \]\[ y = 0 \]So, the y-intercept is at the point \( (0, 0) \).
04

Conclude the Intercepts

Both the x-intercept and the y-intercept are found to be the same point, which is the origin \( (0, 0) \).

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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 linear equation is the point where the graph crosses the x-axis. At this point, the value of y is always zero. To find the x-intercept of the given equation, set y to 0 in the equation and solve for x. For example, in the equation \( x - 5y = 0 \), setting y to 0 gives us \( x - 5(0) = 0 \), which simplifies to \( x = 0 \). So, the x-intercept is the point (0,0).
y-intercept
The y-intercept of a linear equation is the point where the graph crosses the y-axis. At this point, the value of x is always zero. To find the y-intercept, you set x to 0 in the equation and solve for y. For instance, in the equation \( x - 5y = 0 \), setting x to 0 gives \( 0 - 5y = 0 \), which simplifies to \( y = 0 \). Therefore, the y-intercept is also the point (0,0).
linear equations
Linear equations are equations of the first degree, meaning they involve only the first powers of x and y. These equations create a straight line when graphed on a coordinate plane. The standard form of a linear equation is \( ax + by = c \), where a, b, and c are constants. In the given example, \( x - 5y = 0 \) is already in a linear form. Understanding linear equations helps in finding both x and y intercepts since they describe a straight path on a graph.
graphing equations
Graphing linear equations involves plotting points that satisfy the equation and then connecting them to draw a straight line. For accurate graphing, it's important to determine at least two points. Usually, finding the x-intercept and y-intercept is sufficient. As seen in the equation \( x - 5y = 0 \), both intercepts are (0,0). This means the line only crosses the origin. Understanding how to plot the intercepts helps visualize the equation and see how it behaves on a graph.

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