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Chapter 7: Problem 16

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

Short Answer

Expert verified
Both intercepts are at (0, 0). The graph is a straight line through the origin with slope \(\frac{1}{3}\).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y to 0 and solve for x. Given the equation: \[x - 3y = 0\]Set y to 0:\[x - 3(0) = 0\]Then, simplify\[x = 0\]So, the x-intercept is (0, 0).
02

Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. Given the equation: \[x - 3y = 0\]Set x to 0:\[0 - 3y = 0\]Then, simplify:\[ -3y = 0\]Divide both sides by -3:\[ y = 0 \]So, the y-intercept is (0, 0).
03

Graph the equation

The equation can be rewritten as: \[y = \frac{x}{3}\]Since both intercepts are at (0, 0), this is the origin. Plot the origin point and use the slope to find another point. The slope is \(\frac{1}{3}\), meaning when x increases by 3 units, y increases by 1 unit. Plot another point at (3, 1) and draw a straight line through these points to graph 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 in a graph is where the line crosses the x-axis. This point is important because it tells us where the value of y is zero. For our equation, we set y to 0 and solve for x:

Given the equation: \[ x - 3y = 0 \] We set y to 0: \[ x - 3(0) = 0 \] This simplifies to x = 0.

So, the x-intercept is (0, 0). This means that the graph crosses the x-axis at the origin when y is zero.
y-intercept
The y-intercept is the point where the line crosses the y-axis. This occurs when the value of x is zero. To find this for our equation, we set x to 0 and solve for y:

Given the equation: \[ x - 3y = 0 \] We set x to 0: \[ 0 - 3y = 0 \] This simplifies to -3y = 0. Dividing both sides by -3, we get y = 0.

So, the y-intercept is (0, 0). This means the graph crosses the y-axis at the origin when x is zero.
slope
The slope of a line measures its steepness and direction. It's often represented by 'm' in the equation y = mx + b. For our equation, we can rewrite it in the slope-intercept form:

\[ y = \frac{x}{3} \] Here, the slope m is \[ \frac{1}{3} \].

This means for every 3 units we move right on the x-axis, we move 1 unit up on the y-axis.
  • Start by plotting the intercept point at the origin (0,0).
  • Use the slope to find another point: moving 3 units to the right on the x-axis and 1 unit up on the y-axis gives us the point (3,1).
Drawing a line through these points gives us the graph of our equation.

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