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Chapter 2: Problem 78

Write an equation of the line passing through the given point and satisfying the given condition. Give the equation (a) in slope-intercept form and (b) in standard form. See Example 6. $$ (2,-7) ; \text { perpendicular to } 5 x+2 y=18 $$

Short Answer

Expert verified
The equations of the line are: (a) Slope-intercept form: \( y = \frac{2}{5}x - \frac{39}{5} \) (b) Standard form: \( 2x - 5y = 39 \)

Step by step solution

01

Identify the slope of the given line

Rewrite the given line equation in slope-intercept form: \[ 5x + 2y = 18 \] Solving for y gives: \[ 2y = -5x + 18 \] \[ y = -\frac{5}{2}x + 9 \] The slope of the given line is \(-\frac{5}{2}\).
02

Determine the slope of the perpendicular line

The slope of the line perpendicular to \(y = -\frac{5}{2}x + 9\) will be the negative reciprocal of \(-\frac{5}{2}\). So, the slope of the perpendicular line is \(\frac{2}{5}\).
03

Use the point-slope form to find the equation of the perpendicular line

The point-slope form of an equation of a line is given by: \( y - y_1 = m(x - x_1) \). Substituting the point \((2, -7)\) and the slope \(\frac{2}{5}\) into the equation: \[ y + 7 = \frac{2}{5}(x - 2) \] Simplifying this, we get: \[ y + 7 = \frac{2}{5}x - \frac{4}{5} \] \[ y = \frac{2}{5}x - \frac{4}{5} - 7 \] \[ y = \frac{2}{5}x - \frac{4}{5} - \frac{35}{5} \] \[ y = \frac{2}{5}x - \frac{39}{5} \]
04

Write the equation in slope-intercept form

The slope-intercept form of the equation is: \( y = \frac{2}{5}x - \frac{39}{5} \)
05

Convert the equation to standard form

To convert \( y = \frac{2}{5}x - \frac{39}{5} \) to standard form: Multiply every term by 5 to eliminate the fractions: \[ 5y = 2x - 39 \] Rearrange the equation to standard form: \[ 2x - 5y = 39 \]

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Key Concepts

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

slope-intercept form
Let's start with the slope-intercept form. This is a way of writing the equation of a line so it's easy to picture the slope and the y-intercept. The general form is








Whereas, - The constant 'b' indicates where the line crosses the y-axis (the y-intercept). For example, in the equation





At point (2, -7), y = -7 and m = 5/2. Using the point-slope form, we will write the equation as
The slope-intercept form equation is .
standard form
Now, let's move on to standard form. The standard form of a line equation is to rearrange any given line equation to fit this structure. If the simplest way
equation in standard form
<2x_ctis played. 2y = x - 39

make sure
perpendicular lines
Finally! Let's talk about perpendicular lines. form whenever

slope. For example, if slope
.hift-functioned slope-inter }<|vq_3319|>

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Most popular questions from this chapter

A factory can have no more than 200 workers on a shift, but must have at least 100 and must manufacture at least 3000 units at minimum cost. How many workers should be on a shift in order to produce the required units at minimal cost? Let \(x\) represent the number of workers and y represent the number of units manufactured. Write three inequalities expressing the problem conditions.

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