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Chapter 11: Problem 138

In the following exercises, find the intercepts. $$ 3 x+y=9 $$

Short Answer

Expert verified
The y-intercept is (0, 9) and the x-intercept is (3, 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 and solve for y. Equation: \[ 3x + y = 9 \] Set \( x = 0 \): \[ 3(0) + y = 9 \] So, \( y = 9 \). Therefore, the y-intercept is (0, 9).
02

Find the x-intercept

To find the x-intercept, set the value of y to 0 in the equation and solve for x. Equation: \[ 3x + y = 9 \] Set \( y = 0 \): \[ 3x + 0 = 9 \] So, \( 3x = 9 \) Solving for x, we get: \[ x = 3 \] Therefore, the x-intercept is (3, 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 is the point where a graph crosses the y-axis. At this point, the value of x is always zero because you are not moving left or right, just up or down.

To find the y-intercept, set x to zero and solve for y. In the equation \(3x + y = 9\), let's set \(x = 0\):
\(3(0) + y = 9\)
Simplifying this, we get \(y = 9\).
Therefore, the y-intercept is (0, 9). It tells us that when x is zero, y is nine. It provides a crucial point for graphing linear equations and understanding their behavior.
x-intercept
The x-intercept is the point where a graph crosses the x-axis. At this point, the value of y is always zero because you are not moving up or down, just left or right.

To find the x-intercept, set y to zero and solve for x. In the equation \(3x + y = 9\), let's set \(y = 0\):
\(3x + 0 = 9\)
Solving for x, we get \(3x = 9\) which simplifies to \(x = 3\).
Therefore, the x-intercept is (3, 0). This tells us that when y is zero, x is three. The x-intercept helps in plotting the graph of the equation and gives an important insight into its path.
linear equations
Linear equations are equations of the first degree. This means the highest power of the variable is one. They form straight lines when graphed, which is why they are called 'linear'.

A common form of a linear equation is \(y = mx + b\), where:
  • \(m\) represents the slope, showing how steep the line is
  • \(b\) represents the y-intercept, showing where the line crosses the y-axis
Given our equation \(3x + y = 9\), it can be rearranged to fit the form \(y = mx + b\). Rearranging, we get \(y = -3x + 9\), where
  • the slope \(m = -3\)
  • the y-intercept \(b = 9\)
This linear equation tells us a lot about the line鈥檚 direction and placement. Understanding how to work with linear equations is key in algebra.

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Most popular questions from this chapter

In the following exercises, graph by plotting points. $$ 2 x+y=-3 $$

In the following exercises, graph the vertical or horizontal lines. $$ x=3 $$

Highway grade A local road rises 2 feet for every 50 feet of highway. (a) What is the slope of the highway? (b) The grade of a highway is its slope expressed as a percent. What is the grade of this highway?

In the following exercises, graph the vertical and horizontal lines. $$ y=-5 $$

In the following exercises, identify the most convenient method to graph each line. $$ y=-3 $$
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