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Chapter 11: Problem 150

In the following exercises, find the intercepts. $$ y=\frac{1}{3} x+4 $$

Short Answer

Expert verified
The y-intercept is (0, 4) and the x-intercept is (-12, 0).

Step by step solution

01

- Find the y-intercept

To find the y-intercept, set the value of x to 0 in the equation. The equation is: \( y = \frac{1}{3}x + 4 \)Setting \( x = 0 \): \( y = \frac{1}{3}(0) + 4 \)This simplifies to \( y = 4 \). The y-intercept is (0, 4).
02

- Find the x-intercept

To find the x-intercept, set the value of y to 0 in the equation. Starting from the same equation: \( y = \frac{1}{3}x + 4 \)Setting \( y = 0 \): \( 0 = \frac{1}{3}x + 4 \)Subtract 4 from both sides: \( -4 = \frac{1}{3}x \)Multiply both sides by 3: \( -12 = x \)So, the x-intercept is (-12, 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. To find this intercept, we set the value of x to 0 and solve for y. In the given equation, \( y = \frac{1}{3}x + 4 \), substituting \( x = 0 \) simplifies the equation to \( y = 4 \). Therefore, the y-intercept is at the point (0, 4). Remember, the y-intercept always has the format (0, y). This point is crucial because it helps us quickly determine one of the points the line passes through.
x-intercept
The x-intercept is where the line crosses the x-axis, and to find it, we set the value of y to 0 and solve for x. For our equation, \( y = \frac{1}{3}x + 4 \), setting \( y = 0 \) yields \( 0 = \frac{1}{3}x + 4 \). Solving this step-by-step: first subtract 4 from both sides to get \( -4 = \frac{1}{3}x \). Then, multiply both sides by 3 to isolate x, which gives us \( x = -12 \). Hence, the x-intercept is at (-12, 0). Note that the x-intercept is always in the format (x, 0). This point tells us where the line meets the x-axis.
linear equations
Linear equations represent straight lines on a graph and have the form \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept. In our example, \( y = \frac{1}{3}x + 4 \), the slope \( m \) is \( \frac{1}{3} \) and the y-intercept \( b \) is 4. These types of equations are fundamental in algebra because they describe the relationship between two variables, x and y. By knowing how to manipulate and solve these equations, you can understand how changes in one variable affect the other.
solving equations step-by-step
Solving linear equations step-by-step ensures accuracy and clarity. For example, to find the intercepts of \( y = \frac{1}{3}x + 4 \), we undertook the following steps:
  • For the y-intercept, set \( x = 0 \) and solve for y.
  • For the x-intercept, set \( y = 0 \) and solve for x.
These steps can be applied to any linear equation to find its intercepts systematically. Breaking down problems makes them easier to manage and understand, reducing the risk of mistakes and improving your mathematical skills.

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