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Chapter 3: Problem 55

Find the \(x\) - and \(y\) -intercepts for each line and use them to graph the line. $$y=2 x-4$$

Short Answer

Expert verified
The y-intercept is (0, -4) and the x-intercept is (2, 0).

Step by step solution

01

- Find the y-intercept

To find the y-intercept, set the value of x to 0 in the equation and solve for y. \[ y = 2(0) - 4 \] Simplifies to: \[ y = -4 \] Thus, the y-intercept is (0, -4).
02

- Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. \[ 0 = 2x - 4 \] Rearranging gives: \[ 2x = 4 \] Divide both sides by 2: \[ x = 2 \] Thus, the x-intercept is (2, 0).
03

- Graph the line

Plot the intercepts found in Steps 1 and 2 on a coordinate plane. Plot the points (0, -4) and (2, 0). Draw a straight line through these points, which represents the given equation.

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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 line is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero. To find the x-intercept, you follow these steps:

1. Start with the linear equation; for example, in the exercise: \( y = 2x - 4 \).
2. Set \( y = 0 \) because the y-coordinate at the x-intercept is zero.
3. Substitute zero for \( y \) in the equation and solve for \( x \).
4. In our example, we set \( 0 = 2x - 4 \).
5. Solve for \( x \): \( 2x = 4 \).
6. Divide by 2: \( x = 2 \).
So, the x-intercept is at the point \( (2, 0) \).
Plot this point on the graph where \( x = 2 \) and \( y = 0 \).
Understanding the y-intercept
The y-intercept is where the line crosses the y-axis. At this point, the x-coordinate is always zero. To find this:

1. Take the linear equation given; for example: \( y = 2x - 4 \).
2. Set \( x = 0 \) since the x-coordinate at the y-intercept is zero.
3. Substitute zero for \( x \) in the equation and solve for \( y \).
4. In this case, we get: \( y = 2(0) - 4 \).
5. Simplifying gives: \( y = -4 \).
So, the y-intercept is at the point \( (0, -4) \).
Plot this point on the graph where \( x = 0 \) and \( y = -4 \).
Graphing linear equations
Graphing a linear equation involves plotting points and drawing a straight line through them. For the equation \( y = 2x - 4 \), we already found the intercepts:

  • The y-intercept is \( (0, -4) \).
  • The x-intercept is \( (2, 0) \).
Now, follow these steps:

1. Plot the y-intercept (0, -4) on graph paper by locating the point on the y-axis where \( y = -4 \).
2. Plot the x-intercept (2, 0) on the x-axis where \( x = 2 \).
3. Use a ruler to draw a straight line through both points.
This line represents the equation \( y = 2x - 4 \). By connecting the x- and y-intercepts, you can visualize the behavior of the linear equation.

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Most popular questions from this chapter

Determine which of the ordered pairs \((1,3),(-2,5)\) \((-6,-4),\) and \((7,-8)\) satisfy each compound or absolute value inequality. $$y>-x+1 \text { or } y>4 x$$

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If Mildred earns \(\$ 14.50\) per hour, express her total pay \(P(h)\) as a function of the number of hours worked \(h .\) Find \(P(40)\).

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Find all intercepts for each line. Some of these lines have only one intercept. $$2-10 y=0$$
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