Problem 44 Find the $x$ - and $y$ -inte... [FREE SOLUTION]

Chapter 2: Problem 44

Find the $x$ - and $y$ -intercepts. Then graph each equation. $$ x-3 y=0 $$

Short Answer

Expert verified

Both the x- and y-intercepts are (0, 0). Plot the point (0, 0) and draw the line through it.

Step by step solution

Identify the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. Starting with the equation: \[ x - 3y = 0 \ \text{Set } y = 0: \ x - 3(0) = 0 \ x = 0 \ \text{The x-intercept is } (0, 0) \]

Identify the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. Starting with the equation: \[ x - 3y = 0 \ \text{Set } x = 0: \ 0 - 3y = 0 \ -3y = 0 \ y = 0 \ \text{The y-intercept is } (0, 0) \]

Plot the intercepts

Plot the intercepts (0, 0) on the coordinate plane. Because both intercepts are the same point, we only have this one point.

Draw the graph

Since the equation $ x - 3y = 0 $ is a linear equation, it represents a straight line. In this case, the line coincides with the origin and extends in both directions along the line described by the equation.

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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 line crosses the x-axis. To find the x-intercept, we set the value of y to 0 and solve for x. In our example with the equation $ x - 3y = 0 $, setting $ y = 0 $ gives us the equation: $ x - 3(0) = 0 $. Simplifying this, we get $ x = 0 $. Thus, the x-intercept is at the point (0, 0). This point is where the line touches the x-axis.

y-intercept

The y-intercept of a linear equation is the point where the line crosses the y-axis. To find the y-intercept, we set the value of x to 0 and solve for y. For our example with the equation $ x - 3y = 0 $, setting $ x = 0 $ gives us: $ 0 - 3y = 0 $. Simplifying this, we find $ y = 0 $. Hence, the y-intercept is also at the point (0, 0). This point is where the line touches the y-axis.

graphing linear equations

Graphing linear equations involves plotting points on the coordinate plane that satisfy the given equation, and then drawing a line through these points. For the equation $ x - 3y = 0 $, both the x-intercept and y-intercept are at (0, 0). So, we plot this point on the coordinate plane. Since this is our only point of intersection but we know this equation represents a straight line, we understand it must pass through the origin. A line through (0, 0) for this simple equation is a straight line that extends in both directions along the trend determined by our equation.

coordinate plane

The coordinate plane is a two-dimensional surface created by two perpendicular lines: the x-axis, which runs horizontally, and the y-axis, which runs vertically. These axes intersect at a point called the origin, denoted as (0, 0). Each point in this plane can be described by an ordered pair of numbers (x, y). When graphing linear equations like $ x - 3y = 0 $, we use the coordinate plane to mark our intercepts and visualize the line that represents the equation. For our example, both intercepts are at (0, 0), making our task to simply plot the single point where the graph intersects both axes.

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91影视

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ x-3 y=0 $$

Short Answer

Step by step solution

Identify the x-intercept

Identify the y-intercept

Plot the intercepts

Draw the graph

Key Concepts

x-intercept

y-intercept

graphing linear equations

coordinate plane

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