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Chapter 1: Problem 14

Find the equation of the line that passes through the point \(\left(-5, \frac{11}{2}\right)\) and is perpendicular to the line with equation \(2 x-5 y-35=0 .\) Write the line in slope-intercept form \((y=m x+c).\)

Short Answer

Expert verified
The equation is \(y = -\frac{5}{2}x - 7\).

Step by step solution

01

Identify the Slope of Given Line

Start by rearranging the equation of the given line, \(2x - 5y - 35 = 0\), into slope-intercept form \(y = mx + c\). After rearranging, we need to solve for \(y\) to find the slope \(m\).
02

Solve for y in Slope-Intercept Form

The equation is \(2x - 5y = 35\). Solving for \(y\), add \(5y\) to both sides and then subtract \(35\): \(5y = 2x - 35\). Then divide everything by 5 to isolate \(y\): \(y = \frac{2}{5}x - 7\). Now, \(m = \frac{2}{5}\) is the slope of the given line.
03

Find the Perpendicular Slope

The slope of a line perpendicular to another line is the negative reciprocal of the original slope. For a line with slope \(\frac{2}{5}\), the perpendicular slope is \(-\frac{5}{2}\).
04

Use Point-Slope Form to Find the Equation

Using the point \((-5, \frac{11}{2})\) and the slope \(-\frac{5}{2}\), apply the point-slope form of a line equation: \(y - y_1 = m(x - x_1)\). Substitute \(x_1 = -5\), \(y_1 = \frac{11}{2}\), and \(m = -\frac{5}{2}\): \[ y - \frac{11}{2} = -\frac{5}{2}(x + 5) \].
05

Simplify to Slope-Intercept Form

Distribute the slope on the right: \(y - \frac{11}{2} = -\frac{5}{2}x - \frac{25}{2}\). Then add \(\frac{11}{2}\) to both sides to solve for \(y\): \[ y = -\frac{5}{2}x - \frac{25}{2} + \frac{11}{2} \] Simplify: \[ y = -\frac{5}{2}x - \frac{14}{2} \] Finally: \( y = -\frac{5}{2}x - 7\).

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Key Concepts

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

Slope-Intercept Form
Understanding the slope-intercept form of a line is crucial for analyzing linear relationships. In its most widely-known form, it is represented as \( y = mx + c \), where:
  • \( y \) is the dependent variable.
  • \( m \) represents the slope of the line.
  • \( x \) is the independent variable.
  • \( c \) is the y-intercept, the point where the line crosses the y-axis.
To convert an equation into slope-intercept form, isolate \( y \) on one side. This form is beneficial because it easily reveals the slope and y-intercept, allowing for straightforward graphing. It helps in visualizing how changes in \( x \) affect \( y \), using the slope \( m \) as the rate of change. For instance, a slope of \( \frac{2}{5} \) indicates that for every movement of 5 units along the x-axis, \( y \) increases by 2 units if the slope is positive.
Point-Slope Form
The point-slope form is an efficient way to write the equation of a line when you know a specific point on that line and its slope. Expressed as \( y - y_1 = m(x - x_1) \), this form allows you to build the equation of a line directly from:
  • \( (x_1, y_1) \), a specific point on the line.
  • \( m \), the slope of the line.
Using the point \( (-5, \frac{11}{2}) \) and the slope \( -\frac{5}{2} \), as in the original problem, we can form the equation: \[ y - \frac{11}{2} = -\frac{5}{2}(x + 5) \]This approach is particularly useful because it provides a clear link between the slope and specific points, making it simple to see how a known point affects the overall line.
Negative Reciprocal
In geometry, perpendicular lines have a unique relationship between their slopes: the slope of one line is the negative reciprocal of the other. To find the negative reciprocal of a slope, you:
  • Flip the original slope.
  • Change its sign.
For example, if a line has a slope of \( \frac{2}{5} \), a line perpendicular to it will have a slope of \( -\frac{5}{2} \). This is because multiplying the slopes of two perpendicular lines will always give \(-1\). This property is key in problems that involve orthogonality, as it lets us easily determine the slope of a line that is perpendicular. By applying this concept, we could determine the perpendicular slope necessary for solving the original problem.

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Most popular questions from this chapter

Find all values of \(x\) that make the equation true. $$|x|=5$$

Use scientific notation and the laws of exponents to perform the indicated operations. Give the result in scientific notation rounded to two significant figures. $$\frac{4 \times 10^{4}}{\left(6.4 \times 10^{2}\right)\left(2.5 \times 10^{-5}\right)}$$

Find all values of \(x\) that make the equation true. $$2\left|\frac{x+2}{2}\right|=2$$

Write an inequality to represent the given interval and state whether the interval is closed, open or half-open. Also state whether the interval is bounded or unbounded. $$]-\infty, 4]$$

Plot the two real numbers on the real number line, and then find the exact distance between their coordinates. $$\frac{61}{7} \text { and }-\frac{23}{11}$$
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