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Chapter 4: Problem 161

Find the intercepts for each equation. \(3 x+y=6\)

Short Answer

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

Step by step solution

01

- Find the x-intercept

To find the x-intercept, set y = 0 in the equation 3x + y = 6 . Solve for x : 3x + 0 = 6 3x = 6 x = 2 . Thus, the x-intercept is (2, 0) .
02

- Find the y-intercept

To find the y-intercept, set x = 0 in the equation 3x + y = 6 . Solve for y : 3(0) + y = 6 y = 6 . Thus, the y-intercept is (0, 6) .

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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 is where the line crosses the x-axis. At this point, the value of y is always zero. To find the x-intercept, follow these steps:

  • Set y to 0 in your equation.
  • Solve for x.
For example, consider the equation: \( 3x + y = 6 \).

Set y to 0: \( 3x + 0 = 6 \).
Now solve for x: \ 3x = 6 \ x = 2 \.

So, the x-intercept is the point (2, 0). This means the line crosses the x-axis at x = 2.
y-intercept
The y-intercept is where the line crosses the y-axis. At this point, the value of x is always zero. Finding the y-intercept involves the following:

  • Set x to 0 in your equation.
  • Solve for y.
Using the same equation: \( 3x + y = 6 \).

Set x to 0: \( 3(0) + y = 6 \).
Now solve for y: \ y = 6 \.

So, the y-intercept is the point (0, 6). This means the line crosses the y-axis at y = 6.
linear equations
A linear equation is a type of algebraic equation where the highest exponent of any variable is 1. Linear equations can usually be written in the form \( Ax + By = C \) where A, B, and C are constants.

Key features of linear equations include:

  • They graph as straight lines.
  • They have a constant rate of change or slope.
  • They can be used to model real-world situations where the relationship between variables is linear.
In our example, the equation \( 3x + y = 6 \) is a linear equation. It represents a straight line when graphed, and both intercepts (x-intercept of (2, 0) and y-intercept of (0, 6)) help us understand more about the line's behavior.

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Most popular questions from this chapter

Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form. . line \(4 x+5 y=-3\), point (0,0)

Find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form. line \(y=4 x+2,\) point (1,2)

Find the equation of each line. Write the equation in slope-intercept form. Parallel to the line \(2 x+3 y=6\), containing point (0,5)

Graph the linear inequality \(x-y \geq-2\)

The age, \(x,\) and LDL cholesterol level, \(y\), of two men are given by the points (18,68) and (27,122) . Find a linear equation that models the relationship between age and LDL cholesterol level.
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