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Chapter 3: Problem 25

Find the \(x\) - and \(y\) -intercepts for the graph of each equation. $$ y=\frac{1}{3} x-2 $$

Short Answer

Expert verified
The y-intercept is (0, -2) and the x-intercept is (6, 0).

Step by step solution

01

- Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\). This is because the y-intercept is where the graph crosses the y-axis, and at this point, \(x\) is always 0. So, substitute \(x = 0\) into the equation \(y = \frac{1}{3}x - 2\):\[y = \frac{1}{3}(0) - 2\]Simplify:\[y = -2\]Therefore, the y-intercept is at (0, -2).
02

- Find the x-intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\). This is because the x-intercept is where the graph crosses the x-axis, and at this point, \(y\) is always 0. So, substitute \(y = 0\) into the equation \(y = \frac{1}{3}x - 2\):\[0 = \frac{1}{3}x - 2\]Solve for \(x\). First, add 2 to both sides:\[2 = \frac{1}{3}x\]Next, multiply both sides by 3 to isolate \(x\):\[x = 6\]Therefore, the x-intercept is at (6, 0).

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Key Concepts

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

y-intercept
The y-intercept of a linear equation is the point where the line crosses the y-axis. This means that at this point, the value of the x-coordinate is always zero. To find the y-intercept, we set x = 0 in the given equation and then solve for y.
Let's illustrate this with the given linear equation:
  • Start with the equation: \(y = \frac{1}{3}x - 2 \)
  • Set \(x = 0\) because we are trying to find the y-intercept: \(y = \frac{1}{3}(0) - 2\)
  • Simplify to get \(y = -2\)
Therefore, the y-intercept is at the point \((0, -2)\). The process is straightforward but essential for understanding the graph's starting position along the y-axis.
x-intercept
The x-intercept of a linear equation is the point where the line crosses the x-axis. At this location, the y-coordinate is always zero. To find the x-intercept, we set y = 0 and solve for x.
With the given equation, we find the x-intercept as follows:
  • Start with the equation: \(y = \frac{1}{3}x - 2\)
  • Set \(y = 0\) for the x-intercept: \(0 = \frac{1}{3}x - 2\)
  • Rearrange the equation to solve for x: \(2 = \frac{1}{3}x\)
  • Multiply both sides by 3: \(x = 6\)
This means that the x-intercept is at the point \((6, 0)\). Knowing the x-intercept helps in understanding where the line crosses the x-axis and is crucial for plotting the equation.
linear equation
A linear equation describes a straight line on a graph. It is usually written in the form \(y = mx + b\), where m represents the slope, and b represents the y-intercept.
Key aspects of a linear equation include:
  • The **slope (m)** indicates the steepness and direction of the line. A positive slope means the line ascends from left to right, while a negative slope means it descends.
  • The **y-intercept (b)** is where the line crosses the y-axis. It's the value of y when x is 0.
Let's look at our example equation:
  • Equation: \(y = \frac{1}{3}x - 2\)
  • Slope (m): \(\frac{1}{3}\), indicating a gentle rise of the line.
  • y-intercept (b): \(-2\), showing where the line crosses the y-axis.
By understanding these components, you can quickly determine how the line behaves and what it looks like on a graph.

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Most popular questions from this chapter

Write an equation of the line passing through the given pair of points. Give the final answer in (a) slope-intercept form and (b) standard form. (-2,-1) and (3,-4)

Describe what the graph of each linear equation will look like in the coordinate plane. (Hint: Rewrite the equation if necessary so that it is in a more recognizable form.) $$ x-10=1 $$

Describe what the graph of each linear equation will look like in the coordinate plane. (Hint: Rewrite the equation if necessary so that it is in a more recognizable form.) $$ 5 y=-15 $$

$$ y=\frac{7}{3} x-6 $$

Graph each linear equation. $$ y=-1 $$
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