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Chapter 11: Problem 155

In the following exercises, graph using the intercepts. $$ -x+5 y=10 $$

Short Answer

Expert verified
The x-intercept is \((-10, 0\), the y-intercept is \((0, 2\); 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 to 0 in the equation \t\begin{eqnarray}-x + 5(0) &=& 10 \-x &=& 10 \x &=& -10\tt. So, the x-intercept is \((-10, 0\).
02

- Find the y-intercept

To find the y-intercept, set x to 0 in the equation \t\begin{eqnarray}-(0) + 5y &=& 10 \5y &=& 10 \y &=& 2\tt. So, the y-intercept is \((0, 2)\).
03

- Plot the intercepts on the graph

Plot the points \((-10, 0)\) and \((0, 2)\) on the graph. These points are the x- and y-intercepts respectively.
04

- Draw the line

Draw a straight line passing through the points \((-10, 0)\) and \((0, 2)\). This is the graph of the equation \t\begin{eqnarray}-x + 5y = 10.\tt

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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 of a graph is where the line crosses the x-axis. This occurs when the value of y is zero. To find the x-intercept for any equation, you need to set y to 0 and solve for x.

In our example, we start with the equation \( -x + 5y = 10 \). When we set y to 0:

\[ -x + 5(0) = 10 \]

This simplifies to:

\[ -x = 10 \]

By multiplying both sides by -1, we get:

\[ x = -10 \]

So, the x-intercept is the point \( (-10, 0) \). This point tells us that the line touches the x-axis at x = -10.

To find the x-intercept:
  • Set y to 0 in the equation
  • Solve the resulting equation for x
  • The solution is the x-intercept \( (x, 0) \)
Understanding the y-intercept
The y-intercept of a graph is where the line crosses the y-axis. This occurs when the value of x is zero. To find the y-intercept for any equation, you need to set x to 0 and solve for y.

In our example, we start with the equation \[ -x + 5y = 10 \]. When we set x to 0:

\[ -(0) + 5y = 10 \]

This simplifies to:

\[ 5y = 10 \]

By dividing both sides by 5, we get:

\[ y = 2 \]

So, the y-intercept is the point \[ (0, 2) \]. This point tells us that the line touches the y-axis at y = 2.

To find the y-intercept:
  • Set x to 0 in the equation
  • Solve the resulting equation for y
  • This solution is the y-intercept \( (0, y) \)
Plotting Points on a Graph
One of the fundamental steps to graphing linear equations is plotting points. Each point can be represented as (x, y) and shows a specific location on the graph.

To graph an equation using intercepts, follow these steps:
  • First, find the x-intercept and y-intercept as discussed.
  • Next, plot these points on the graph. For our equation \[ -x + 5y = 10 \], you will plot points \[ (-10, 0) \] and \[ (0, 2) \].
  • Mark these points clearly. You can use any different symbol or color if it helps.
  • Finally, draw a straight line through these points. Make sure the line extends in both directions beyond these points to indicate it continues indefinitely.


Always recheck to ensure the plotted points match the intercepts you calculated. This step confirms accuracy before you proceed to draw the line.

Tips for plotting points:
  • Use graph paper for precision.
  • Label your axes and scale appropriately.
  • Double-check coordinates before marking.

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