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Chapter 3: Problem 38

In the following exercises, find the intercepts for each equation.. $$ x-y=-4 $$

Short Answer

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

Step by step solution

01

Understand what intercepts are

Intercepts are the points where the graph of an equation crosses the x-axis (x-intercept) and y-axis (y-intercept).
02

Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x.\( x - y = -4 \)Set \( y = 0 \):\( x - 0 = -4 \)So, \( x = -4 \).The x-intercept is \( (-4, 0) \).
03

Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y.\( x - y = -4 \)Set \( x = 0 \):\( 0 - y = -4 \)So, \( -y = -4 \) which means \( y = 4 \).The y-intercept is \( (0, 4) \).

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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 graph is where the graph crosses the x-axis.
This means that at this point, the value of y is 0.
To find the x-intercept of an equation, you set y to 0 and solve for x.
For example, in the equation \(x - y = -4\), you set y to 0, which simplifies the equation to \(x - 0 = -4\).
When solving this, we get \(x = -4\).
So, the x-intercept is \(-4, 0\).
Once you understand this process, you can find the x-intercept for any linear equation by following these steps. This concept is essential because it helps you visualize where the line crosses the x-axis, giving you a starting point for graphing the equation.
y-intercept
The y-intercept of a graph is where the graph crosses the y-axis.
At this point, the value of x is 0.
To find the y-intercept of an equation, you set x to 0 and solve for y.
Using the equation \(x - y = -4\) as an example, you set x to 0, which makes the equation \(0 - y = -4\).
Solving this gives \(-y = -4\), and further solving for y, we find \(y = 4\).
Thus, the y-intercept is \(0, 4\).
Understanding how to find the y-intercept allows you to know where the line crosses the y-axis, assisting in correctly plotting and sketching the graph of the equation.
solving linear equations
Solving linear equations involves finding the values of variables that make the equation true.
For an equation like \(x - y = -4\), it's usually in the form of \(Ax + By = C\).
To solve for one variable, you often simplify the equation based on given conditions, such as setting one variable to zero to find intercepts.
Here are steps to approach solving a linear equation:
  • Isolate the variable you need to solve for by performing algebraic operations.
  • If solving for x, move the y-term to the other side of the equation (or vice versa).
  • Check your solution by plugging it back into the original equation to ensure it satisfies the equation.
Understanding the methods to solve linear equations is fundamental in algebra, as it aids in solving more complex problems and graphical interpretations like finding intercepts.

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