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Chapter 3: Problem 27

Find the \(x\) - and \(y\) -intercepts for the graph of each equation. $$ 2 x-3 y=0 $$

Short Answer

Expert verified
Both the x-intercept and the y-intercept are (0, 0).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set the value of y to 0 and solve for x. Plug y = 0 into the equation: \[ 2x - 3(0) = 0 \] Simplify the equation: \[ 2x = 0 \] Divide both sides by 2: \[ x = 0 \] So, the x-intercept is (0, 0).
02

Find the y-intercept

To find the y-intercept, set the value of x to 0 and solve for y. Plug x = 0 into the equation: \[ 2(0) - 3y = 0 \] Simplify the equation: \[ -3y = 0 \] Divide both sides by -3: \[ y = 0 \] So, the y-intercept is (0, 0).

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Key Concepts

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

finding intercepts
One of the fundamental concepts in algebra is finding intercepts of a line. These intercepts help us understand where a line crosses the axes on a coordinate plane. When given an equation, such as the one in this exercise, we often need to find both the x-intercept and the y-intercept.
To find the intercepts, follow these steps:
  • **Find the x-intercept:** Set y to 0 and solve the equation for x.
  • **Find the y-intercept:** Set x to 0 and solve the equation for y.

Example:
For the equation 2 x - 3 y = 0,
Step-by-step solution:
1. To find the x-intercept, set y to 0 and solve for x: 2 x = 0 x = 0 2. To find the y-intercept, set x to 0 and solve for y: -3 y = 0 y = 0.
In this case, both intercepts are at the origin (0, 0).
linear equations
A linear equation is an equation that forms a straight line when graphed on a coordinate plane. This form of the equation is typically written as: y = mx + b, where m is the slope and b is the y-intercept.
Linear equations can also be written in other forms, such as the standard form Ax + By = C, like the equation: 2x - 3y = 0.
  • A linear equation means the variables x and y are of degree 1.
  • Any changes in x result in linear changes in y and vice versa.
  • Linear equations are easy to solve and graph.

Understanding the properties of linear equations is essential as it helps you find intercepts more efficiently and graph lines accurately.
coordinate plane
The coordinate plane is a two-dimensional plane formed by the intersection of a vertical y-axis and a horizontal x-axis. Each point on this plane is defined by an ordered pair (x, y).
When graphing equations or finding intercepts, the coordinate plane serves as a visual tool.
  • The x-axis runs horizontally and represents the value of x in the coordinates.
  • The y-axis runs vertically and represents the value of y in the coordinates.
  • Wherever a line touches the x-axis is the x-intercept, and wherever it touches the y-axis is the y-intercept.

For the equation 2x - 3y = 0, the intercepts both lie at the origin (0, 0), which is the point where the x-axis and y-axis intersect.

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Most popular questions from this chapter

\(x-5=0\)

Find the \(x\) - and \(y\) -intercepts for the graph of each equation. $$ y=\frac{1}{4} x-1 $$

Graph each linear equation. $$ y=3 $$

Write an equation of the line satisfying the given conditions. Give the final answer in slope-intercept form. (Hint: Recall the relationships among slopes of parallel and perpendicular lines.) Perpendicular to \(x-2 y=7\) \(y\) -intercept (0,-3)

Graph each linear equation. $$ y=2 x-5 $$
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