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Chapter 2: Problem 49

Find the intercepts. Then graph by using the intercepts, if possible, and a third point as a check. $$ x+y=4 $$

Short Answer

Expert verified
The intercepts are (4,0) and (0,4). The third point is (2,2).

Step by step solution

01

Identify x-intercept

To find the x-intercept, set y to 0 in the equation \(x + y = 4\) and solve for x.
02

Solve for x

Substitute \(y = 0\) into the equation: \(x + 0 = 4\). Thus, \(x = 4\). The x-intercept is \((4,0)\).
03

Identify y-intercept

To find the y-intercept, set x to 0 in the equation \(x + y = 4\) and solve for y.
04

Solve for y

Substitute \(x = 0\) into the equation: \(0 + y = 4\). Thus, \(y = 4\). The y-intercept is \((0,4)\).
05

Choose a third point

Select another value for x, for example, \(x = 2\), and solve for y.
06

Solve for y with x = 2

Substitute \(x = 2\) into the equation: \(2 + y = 4\). Solving for y gives \(y = 2\). The third point is \((2,2)\).
07

Graph the points

Plot the x-intercept \((4,0)\), the y-intercept \((0,4)\), and the third point \((2,2)\) on the coordinate plane. Draw a straight line through these three points to graph the equation.

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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 is where a graph crosses the x-axis. This means the value of y at this point is 0. To find the x-intercept of an equation, you need to set y equal to 0 and solve for x.

For the equation given, let's find the x-intercept of the equation \(x + y = 4\).
  • Set y = 0 in the equation: \(x + 0 = 4\).
  • This simplifies to \(x = 4\).
  • Therefore, the x-intercept is (4, 0).
So, whenever you want to find the x-intercept, just set y to 0 and solve for x. Easy peasy! This intercept gives you one point that you can use to graph the equation.
y-intercept
The y-intercept is where a graph crosses the y-axis. This means the value of x at this point is 0. To find the y-intercept of an equation, you need to set x equal to 0 and solve for y.

Let's find the y-intercept of the equation \(x + y = 4\).
  • Set x = 0 in the equation: \(0 + y = 4\).
  • This simplifies to \(y = 4\).
  • Therefore, the y-intercept is (0, 4).
So, by setting x to 0 and solving for y, you find the y-intercept. This helps you get another point to graph your equation.
graphing linear equations
Graphing a linear equation helps you see the relationship between the variables. To graph linear equations efficiently, follow these steps:

1. **Find the intercepts**:
  • Find the x-intercept by setting y = 0 and solving for x. For the equation \(x + y = 4\), the x-intercept is (4, 0).
  • Find the y-intercept by setting x = 0 and solving for y. For the same equation, the y-intercept is (0, 4).

2. **Choose a third point**:
  • Select a different x value and solve for y. For example, if x = 2, then \(2 + y = 4\) simplifies to y = 2. So, (2, 2) is your third point.

3. **Plot the points**:
  • Plot the x-intercept (4, 0), the y-intercept (0, 4), and the additional point (2, 2) on the graph paper.

4. **Draw the line**:
  • Use a ruler to draw a straight line through these points. This line represents the linear equation \(x + y = 4\).

Following these steps will help you graph linear equations quickly and accurately. The intercepts are crucial as they give the easiest points where the line crosses the axes.

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Most popular questions from this chapter

Assume that \(r, p,\) and \(s\) are nonzero constants and that \(x\) and \(y\) are variables. Determine whether each equation is linear. $$ r^{2} x=p y+5 $$

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