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Chapter 4: Problem 26

The \(x\) -intercept is the \(x\) value at which a line crosses the \(x\) -axis. Find an equation of the line with the given \(x\) -intercept and slope. \(x\) -intercept \(5,\) slope \(-2\)

Short Answer

Expert verified
The equation of the line is \( y = -2x + 10 \).

Step by step solution

01

Identify the Point

The line crosses the x-axis at the x-intercept. This gives the point (5, 0) because the y-coordinate is zero at the x-axis.
02

Use Point-Slope Form

The point-slope form of a line's equation is given by \[ y - y_1 = m(x - x_1) \] where \( m \) is the slope, and \( (x_1, y_1) \) is a point on the line. Plugging in \( m = -2 \) and \( (x_1, y_1) = (5, 0) \), the equation becomes \[ y - 0 = -2(x - 5) \].
03

Simplify the Equation

Simplify the equation to get it into the slope-intercept form \( y = mx + b \). Distribute the slope on the right side: \[ y = -2x + 10 \].

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Key Concepts

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

slope-intercept form
The slope-intercept form of a line is one of the most common ways to express the equation of a line. In this form, the equation is written as: \[ y = mx + b \] where:
  • m is the slope of the line
  • b is the y-intercept, the point where the line crosses the y-axis
In the exercise provided, you end up with the equation \( y = -2x + 10 \). Here:
  • The slope m is \textendash2.
  • The y-intercept b is 10, meaning the line crosses the y-axis at (0, 10).
Understanding this form helps in quickly identifying the slope and y-intercept directly from the equation.
point-slope form
The point-slope form of a line is another way to write the equation of a line. It is especially useful when you know a point on the line and the slope. The form is given by: \[ y - y_1 = m(x - x_1) \] where:
  • m is the slope of the line
  • (x_1, y_1) is a specific point on the line
In the exercise, the line passes through the x-intercept (5, 0) and has a slope of \textendash2. Plugging these values in, we get: \[ y - 0 = -2(x - 5) \]. This highlights the point-slope form's utility in converting specific cases to a standard format.
equation of a line
The equation of a line mathematically represents all points that lie on the line. There are multiple forms to express this equation:
  • Slope-intercept form (\[ y = mx + b \])
  • Point-slope form (\[ y - y_1 = m(x - x_1) \])
  • Standard form (\[ Ax + By = C \])
In this exercise, we moved from the point-slope form to the slope-intercept form. The final equation, \[ y = -2x + 10 \]) is the slope-intercept form, readily showing the slope and y-intercept.By understanding these forms, you can convert between them and use the most appropriate one for a given problem.
slope
The slope of a line measures its steepness and the direction in which it increases or decreases. Mathematically, it is the ratio of the change in the y-coordinates to the change in the x-coordinates. It can be calculated using:
  • m = \[ \frac{y_2 - y_1}{x_2 - x_1} \]
In our example, the slope is given as \textendash2. This negative slope means the line decreases as it moves from left to right.Understanding the slope is crucial for drawing lines accurately and interpreting their behavior.

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