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Chapter 3: Problem 56

(a) find the y-intercept. (b) find the x-intercept. (c) find a third solution of the equation. (d) graph the equation. \(x+y=-2000\)

Short Answer

Expert verified
The y-intercept is -2000, the x-intercept is -2000, and a third solution is (-1000, -1000). Plot these points and draw a line through them.

Step by step solution

01

- Find the y-intercept

To find the y-intercept, set the value of x to 0 in the equation. This gives us:\[0 + y = -2000\]Simplify to find y:\[y = -2000\]
02

- Find the x-intercept

To find the x-intercept, set the value of y to 0 in the equation. This gives us:\[x + 0 = -2000\]Simplify to find x:\[x = -2000\]
03

- Find a third solution

A third point can be found by selecting an arbitrary value for x or y. Suppose we choose x = -1000 and substitute it into the equation:\[-1000 + y = -2000\]Solving for y, we get:\[y = -2000 + 1000\]\[y = -1000\]So, a third solution is \((-1000, -1000)\).
04

- Graph the equation

Plot the points found in the previous steps to graph the equation. The points are (0, -2000), (-2000, 0), and (-1000, -1000). Draw a straight line passing through these points to represent the equation \(x + y = -2000\).

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Key Concepts

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

y-intercept
The y-intercept is where the line crosses the y-axis. This means the value of x is 0 at the y-intercept. To find the y-intercept of an equation, substitute 0 for x and solve for y.
In the equation \(x + y = -2000\), set \(x = 0\):

\[0 + y = -2000\]
\[y = -2000\]

The y-intercept is \((0, -2000)\). This point tells us that when x is 0, y is -2000.
This is a crucial point for graphing the equation since it indicates where the line will intersect the y-axis.
x-intercept
The x-intercept is where the line crosses the x-axis, meaning the value of y is 0 at the x-intercept. To find the x-intercept, substitute 0 for y and solve for x.
In the equation \(x + y = -2000\), set \(y = 0\):

\[x + 0 = -2000\]
\[x = -2000\]

The x-intercept is \((-2000, 0)\). This point tells us that when y is 0, x is -2000.
Along with the y-intercept, the x-intercept helps in graphing because it shows where the line crosses the x-axis.
graphing linear equations
Graphing linear equations involves plotting points and drawing a line through them. For the equation \(x + y = -2000\), we have the y-intercept \((0, -2000)\) and the x-intercept \((-2000, 0)\).
Let's now plot these points:

  • Point 1: (0, -2000)
  • Point 2: (-2000, 0)

We also found a third solution in the steps provided: \((-1000, -1000)\). This point lies on the line.
Plot this third point too:

  • Point 3: (-1000, -1000)

Now, draw a straight line passing through these three points. Ensure the line extends in both directions. This visual representation confirms the solutions we calculated.
solving linear equations
Solving linear equations like \(x + y = -2000\) involves finding different values of x and y that satisfy the equation. Here are the steps:
  • Step 1: Set one variable to 0 to find the intercept (y-intercept and x-intercept).
  • Step 2: Solve for the other variable.

1. For y-intercept:

Set \(x = 0\), find \(y\) as:
\[0 + y = -2000\]
\[y = -2000\]

2. For x-intercept:

Set \(y = 0\), find \(x\) as:
\[x + 0 = -2000\]
\[x = -2000\]

3. To find a third solution, pick an arbitrary value for x or y. Suppose \(x = -1000 \):
\[-1000 + y = -2000\]
\[y = -1000\]

Lastly, you graph these solutions to confirm their validity visually.

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Most popular questions from this chapter

Use the slope formula to find the slope of the line that passes through the points. \((-3,8) ;(11,23)\)

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((1,5) ;(3,-3)\)

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(x-3 y=27\)

(a) find three solutions of the equation. (b) graph the equation. \(y=\frac{2}{5} x-3\)

Use the slope formula to find the slope of the line that passes through the points. \((-30,55) ;(-5,80)\)
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