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Chapter 3: Problem 72

Write an equation for each line passing through the given pair of points. Give the final answer in (a) slope-intercept form and (b) standard form. $$ \left(\frac{1}{2}, \frac{3}{2}\right) \text { and }\left(-\frac{1}{4}, \frac{5}{4}\right) $$

Short Answer

Expert verified
(a) y = \(\frac{1}{3}x + \frac{4}{3}\), (b) -x + 3y = 4

Step by step solution

01

Calculate the Slope

Use the slope formula to find the slope (m) of the line passing through the points \(\frac{1}{2}, \frac{3}{2}\) and \(-\frac{1}{4}, \frac{5}{4}\). The slope formula is \(\frac{y_2 - y_1}{x_2 - x_1}\). Substitute the given points into the formula: \(\frac{\frac{5}{4} - \frac{3}{2}}{-\frac{1}{4} - \frac{1}{2}}\).
02

Simplify the Slope

Simplify the expression obtained: \(\frac{\frac{5}{4} - \frac{6}{4}}{-\frac{1}{4} - \frac{2}{4}} = \frac{-\frac{1}{4}}{-\frac{3}{4}} = \frac{1}{3}\). The slope (m) is \(\frac{1}{3}\).
03

Use the Point-Slope Form

Use the point-slope form of a line equation \y - y_1 = m(x - x_1)\. Choose one of the points, say \(\frac{1}{2}, \frac{3}{2}\), and plug it into the equation together with the slope: \y - \frac{3}{2} = \frac{1}{3}(x - \frac{1}{2})\.
04

Convert to Slope-Intercept Form

Simplify the equation to get it into slope-intercept form \(y = mx + b\): \y - \frac{3}{2} = \frac{1}{3}x - \frac{1}{6}\. Add \(\frac{3}{2}\) to both sides: \y = \frac{1}{3}x + \frac{8}{6} = \frac{1}{3}x + \frac{4}{3}\.
05

Convert to Standard Form

Convert the slope-intercept form into standard form \(Ax + By = C\): Multiply everything by 3 to clear the fraction: \3y = x + 4\. Rearrange to standard form: \-x + 3y = 4\.

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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 linear equation is one of the most common ways to represent a line. It is written as:

standard form
In mathematics, the standard form of a linear equation offers a standardized way to write the equation of a line. It is given by:

point-slope form
The point-slope form is extremely helpful when you know a point on the line and the slope of the line. It is structured as:

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Most popular questions from this chapter

Write an equation for each line passing through the given point and having the given slope. Give the final answer in slope-intercept form. $$ (-4,1), m=\frac{3}{4} $$

If two lines are both vertical or both horizontal, which of the following are they? A. Parallel B. Perpendicular C. Neither parallel nor perpendicular

A point \((x, y)\) has the property that \(x y>0 .\) In which quadrant(s) must the point lie? Explain.

Evaluate each expression. $$ \frac{5}{8} \cdot \frac{5}{8} $$

Concept Check Plot each set of points, and draw a line through them. Then give the equation of the line. $$ (-5,5),(0,5), \text { and }(3,5) $$
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