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Chapter 3: Problem 20

Use intercepts and a checkpoint to graph each equation. $$x+y=6$$

Short Answer

Expert verified
The graph of the equation \( x + y = 6 \) would intercept the x-axis at the point (6,0), the y-axis at point (0,6) and it would also pass through the checkpoint (3,3).

Step by step solution

01

Identifying the x-intercept

To find the x-intercept, set y equal to zero and solve for x. That's because the x-intercept is the point where the graph crosses the x-axis, which is when \( y = 0 \). Hence, if \( y = 0 \), then the equation \( x + 0 = 6 \) simplifies to \( x = 6 \). Therefore, the x-intercept is at the point (6,0).
02

Identifying the y-intercept

To find the y-intercept, set x equal to zero and solve for y. This is because the y-intercept is the point where the graph crosses the y-axis, which is when \( x = 0 \). Hence, If \( x = 0 \), then the equation \( 0 + y = 6 \) simplifies to \( y = 6 \). Therefore, the y-intercept is at the point (0, 6).
03

Selecting a suitable checkpoint

A checkpoint is a point on the grid that is used to check the accuracy of the graph. A good checkpoint for this equation would be the point (3, 3) because if you substitute these values into the equation, it holds true: \( 3 + 3 = 6 \).
04

Drawing the graph

Plot the x and y intercepts and the checkpoint on the graph paper. Then, draw a straight line through these points. The line represents all the possible solutions to the equation.

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Key Concepts

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

Intercepts
Intercepts are crucial in graphing linear equations. They help in quickly sketching the graph by providing key points where the line crosses the axes. In our example, the equation is given as \(x + y = 6\).
  • The x-intercept indicates where the line crosses the x-axis and is found by setting \(y = 0\).
  • The y-intercept indicates where the line crosses the y-axis and is found by setting \(x = 0\).
By identifying these intercepts, you have a good basis for graphing the entire line.
To find each intercept, simply substitute and solve within the equation, making the process efficient and straightforward.
x-intercept
The x-intercept is a fundamental aspect of plotting linear equations. By definition, it is the point where the line intersects the x-axis, meaning the value of \(y\) is zero at this point.
Let's find the x-intercept for the equation \(x + y = 6\). Set \(y = 0\) and solve for \(x\):
\[ x + 0 = 6 \] Thus:
\[ x = 6 \]This calculation reveals that the x-intercept occurs at the coordinate point (6, 0). Plot this point on the graph, as it plays a key role in defining the line trajectory.
y-intercept
The y-intercept of a line is equally important as the x-intercept when graphing linear equations. It is the point where the line crosses the y-axis, which means \(x\) must be zero at this location.
For the equation \(x + y = 6\), set \(x = 0\) and solve for \(y\):
\[ 0 + y = 6 \]It leads to:
\[ y = 6 \] Thus, the y-intercept is at (0, 6).
Plot this on the graph, as it's a critical point that, along with the x-intercept, helps define the slope and position of the entire line.
Checkpoint Method
The checkpoint method is an additional step to verify the correctness of your graph. Once you have plotted the intercepts, selecting a third point ensures that the line is accurate. This is done by choosing any point that satisfies the equation and checking if it aligns with the graph.
For the equation \(x + y = 6\), a suitable checkpoint is (3, 3). Substitute these coordinates into the equation:
\[ 3 + 3 = 6 \]Since the equation holds true, (3, 3) is indeed on the line.
Add this point to your graph to form a robust line spanning through all key points, reassuring the accuracy of the drawn line.

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Most popular questions from this chapter

Use a graphing utility to graph each equation in Exercises in a standard viewing rectangle, \([-10,10,1]\) by \([-10,10,1] .\) Then use the \([\text { TRACE }]\) feature to trace along the line and find the coordinates of two points. $$y=2 x-1$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I use \(y=m x+b\) to write equations of lines passing through two points when neither contains the \(y\) -intercept.

Will help you prepare for the material covered in the next section. Remember that a solution of an equation in two variables is an ordered pair. Let \(x=0\) and find a solution of \(3 x-4 y=24\)

A new car worth 45,000 dollars is depreciating in value by 5000 dollars per year. The mathematical model $$y=-5000 x+45,000$$ describes the car's value, \(y,\) in dollars, after \(x\) years. a. Find the \(x\) -intercept. Describe what this means in terms of the car's value. b. Find the \(y\) -intercept. Describe what this means in terms of the car's value. c. Use the intercepts to graph the linear equation. Because \(x\) and \(y\) must be nonnegative (why?), limit your graph to quadrant I and its boundaries. d. Use your graph to estimate the car's value after five years.

Use a graphing utility to graph each equation in Exercises in a standard viewing rectangle, \([-10,10,1]\) by \([-10,10,1] .\) Then use the \([\text { TRACE }]\) feature to trace along the line and find the coordinates of two points. $$y=-3 x+2$$
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