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Chapter 3: Problem 29

Find the intercepts and graph them. $$ 4 x-y=-8 $$

Short Answer

Expert verified
The x-intercept is \((-2,0)\) and the y-intercept is \((0,8)\). Graph these points and draw a line through them.

Step by step solution

01

Identify the equation format

The equation given is \(4x - y = -8\). This equation is in standard form \(Ax + By = C\). The coefficients are \(A = 4\), \(B = -1\), and \(C = -8\).
02

Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\).\[4x - 0 = -8 \4x = -8 \x = \frac{-8}{4} = -2\]The x-intercept is \((-2, 0)\).
03

Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\).\[4(0) - y = -8 \-y = -8 \y = 8\]The y-intercept is \((0, 8)\).
04

Graph the intercepts

Plot the x-intercept \((-2, 0)\) and the y-intercept \((0, 8)\) on a coordinate plane. Draw a line through these points to represent the equation \(4x - y = -8\). This line extends in both directions beyond the intercepts.

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Key Concepts

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

Understanding the x-intercept
The x-intercept is a crucial point of a line on a graph, where the line crosses the x-axis. This means that at the x-intercept, the value of y is zero. To find the x-intercept of any equation, we replace the variable y with zero and then solve for x.
This process can be broken down as follows:
  • Replace y with zero in the linear equation.
  • Solve the resulting equation for x.
  • The solution gives you the x-coordinate of the x-intercept, while the y-coordinate remains zero.
In this exercise, the equation is given as \(4x - y = -8\). By setting \(y = 0\), the equation simplifies to \(4x = -8\). Solving this equation results in \(x = -2\).
Hence, the x-intercept is the point \((-2, 0)\). Finding the x-intercept this way is valuable for sketching graphs, as it provides a specific point through which the line passes.
Understanding the y-intercept
The y-intercept is the other key point on a graph where the line crosses the y-axis. Here, the x-coordinate must be zero. To find the y-intercept, simply substitute x with zero and solve for y in the equation.
  • Set x equal to zero in the linear equation.
  • Solve for the value of y.
  • You'll obtain the y-intercept with coordinates (0, y).
Returning to our equation, \(4x - y = -8\), setting \(x = 0\) leads to \(-y = -8\). Solving this gives \(y = 8\).
Therefore, the y-intercept is at the point \((0, 8)\). Recognizing y-intercepts is fundamental in graphing equations, as it offers another precise point for the line to traverse.
Understanding the standard form
Linear equations are often expressed in multiple formats, and the standard form is one of the most common. The standard form of a linear equation is \(Ax + By = C\), where A, B, and C are real numbers, and A and B are not both zero.
The benefits of using the standard form include:
  • Simplicity in finding both x-intercepts and y-intercepts by substituting zero for either variable.
  • Easy comparison of different linear equations.
  • Clear visibility of the coefficients of x and y.
In our example, the equation \(4x - y = -8\) is already in standard form with coefficient values: \(A = 4\), \(B = -1\), and \(C = -8\). This form allows for straightforward identification of intercepts by setting either x or y to zero and solving. It's a flexible representation that facilitates the understanding and manipulation of linear relationships.

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