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Chapter 3: Problem 45

In the following exercises, graph using the intercepts. $$ -x+4 y=8 $$

Short Answer

Expert verified
The intercepts are \((-8, 0)\) and \((0, 2)\). Plot these points and draw the line through them.

Step by step solution

01

Find the x-intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\). Substitute \(y = 0\) into the equation:\[-x + 4(0) = 8\]This simplifies to \[-x = 8\], so the x-intercept is \(x = -8\). The x-intercept is the point \((-8, 0)\).
02

Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\). Substitute \(x = 0\) into the equation:\[-0 + 4y = 8\]This simplifies to \[4y = 8\], so \(y = 2\). The y-intercept is the point \((0, 2)\).
03

Plot the intercepts on a graph

Plot the points \((-8, 0)\) and \((0, 2)\) on the graph. These are the x-intercept and y-intercept, respectively.
04

Draw the line

Draw a straight line through the points \((-8, 0)\) and \((0, 2)\). This line represents the graph of the equation \(-x + 4y = 8\).

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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 linear equation is the point where the graph crosses the x-axis. To find this point, we set the value of y to zero in the equation and solve for x. For the equation \(-x + 4y = 8\), setting \y = 0\ gives us \(-x + 4(0) = 8\). Simplifying this, we get \-x = 8\, which leads to \(x = -8\). Thus, the x-intercept is at the point \(-8, 0\). Remember the x-intercept always has the form \ (x, 0)\.

To sum up:
  • Set \y = 0\
  • Solve for \x\
  • The x-intercept is \ (x, 0)\
y-intercept
The y-intercept of a linear equation is the point where the graph crosses the y-axis. To find this point, we set the value of x to zero in the equation and solve for y. In the given equation, we set \x = 0\, so it becomes \(-0 + 4y = 8\). Simplifying this, we get \4y = 8\, which means \(y = 2\). Thus, the y-intercept is at the point \(0, 2\). The y-intercept always has the form \ (0, y)\.

Steps for finding the y-intercept:
  • Set \x = 0\
  • Solve for \y\
  • The y-intercept is \ (0, y)\
plotting points
Plotting points is a crucial step in graphing linear equations. Once you have found the intercepts or other important points, you need to mark them on a graph. For the equation \(-x + 4y = 8\), the intercepts are \(-8, 0\) and \(0, 2\). To plot these points:
1. Find the coordinate \(-8\) on the x-axis and mark the point \(-8, 0\).
2. Find the coordinate \2\ on the y-axis and mark the point \(0, 2\).
Once the points are marked, you have the necessary reference points to draw the graph of the equation. This makes it easier to see the relationship between x and y values on the graph.

Remember:
  • Plot x-intercept on the x-axis
  • Plot y-intercept on the y-axis
  • Ensure points are accurately placed
solving equations
Solving equations is essential for finding intercepts and plotting points. It involves isolating the variable you are solving for. For the given equation, \(-x + 4y = 8\), we used simple algebraic operations:

1. Set one of the variables to zero to find the intercept.
2. Solve for the other variable by simplifying the equation.

For instance, to find the x-intercept:
\begin{align*}-x + 4(0) &= 8\-x &= 8\x &= -8\right
To find the y-intercept:
\begin{align*}-0 + 4y &= 8\4y &= 8\y &= 2\right

Understanding these operations and steps is key to graphing linear equations efficiently. Solving for one variable at a time makes complex problems manageable and clear.

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