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Chapter 2: Problem 2

For the line \(2 x+3 y=6,\) the \(x\) -intercept is ______ and the \(y\) -intercept is ____.

Short Answer

Expert verified
The x-intercept is (3, 0) and the y-intercept is (0, 2).

Step by step solution

01

Understand the Equation

The given equation of the line is in standard form: y = 2x + 3y = 6.
02

Find the X-Intercept

To find the x-intercept, set y = 0 in the equation:\[ 2x + 3(0) = 6 \]Solve for x:\[ 2x = 6 \]\[ x = 3 \].So, the x-intercept is (3, 0).
03

Find the Y-Intercept

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

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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 of a line in a coordinate system is the point where the line crosses the x-axis. To find the x-intercept, we set y to zero in the line's equation and solve for x. This is because at any point on the x-axis, the y-coordinate is always zero.

For example, in the equation given in the exercise, \(2x + 3y = 6\), setting \(y = 0\) gives:\begin{align*} \ 2x + 3(0) = 6 \ 2x = 6 \ x = 3 \ \end{align*}
So, the x-intercept is (3, 0). This means that the line crosses the x-axis at the point where x is 3 and y is 0. Simple steps like these can always help you identify the x-intercept easily.
y-intercept
The y-intercept is where the line crosses the y-axis. To find the y-intercept, we set x to zero in the line's equation and solve for y. This is because at any point on the y-axis, the x-coordinate is always zero.

Using the equation from our exercise, \(2x + 3y = 6\), setting \(x = 0\) gives:\begin{align*} \ 2(0) + 3y = 6 \ 3y = 6 \ y = 2 \ \end{align*}
So, the y-intercept is (0, 2). This means that the line intersects the y-axis at the point where y is 2 and x is 0. Finding the y-intercept follows a similar approach to finding the x-intercept, and practicing these steps will solidify your understanding.
standard form of a line
The standard form of a line is a way of writing the equation of a line so that it is clear and organized. It is generally written as \(Ax + By = C\), where A, B, and C are integers, and A should be a non-negative integer.

In our exercise, the equation \(2x + 3y = 6\) is already in standard form. Here:
  • A = 2
  • B = 3
  • C = 6

The standard form is helpful because it makes it easier to find both the x-intercept and the y-intercept directly. It organizes the equation in a form where algebraic manipulation and understanding intercepts become straightforward. Mastering how to work with this form will enable you to handle more complex linear equations effectively.

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Most popular questions from this chapter

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. A square with an area of 64 square meters is inscribed inside a circle. What are the circumference and area of the circle?

(a) find the intercepts of the graph of each equation and (b) graph the equation. $$ 7 x+2 y=21 $$

The stress in the material of a pipe subject to internal pressure varies directly with the internal pressure and the internal diameter of the pipe and inversely with the thickness of the pipe. The stress is 100 pounds per square inch when the diameter is 5 inches, the thickness is 0.75 inch, and the internal pressure is 25 pounds per square inch. Find the stress when the internal pressure is 40 pounds per square inch if the diameter is 8 inches and the thickness is 0.50 inch.

Find an equation for the line with the given properties. Express your answer using either the general form or the slope-intercept form of the equation of a line, whichever you prefer. Perpendicular to the line \(y=8\); containing the point (3,4)

(a) find the intercepts of the graph of each equation and (b) graph the equation. $$ 0.2 x-0.5 y=1 $$
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