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Chapter 1: Problem 34

Find an equation of the line that passes through the given points. $$ (-1,-2) \text { and }(3,-4) $$

Short Answer

Expert verified
The equation of the line passing through the points (-1, -2) and (3, -4) is \(y = -\frac{1}{2}x - \frac{5}{2}\).

Step by step solution

01

Determine the Slope

To find the slope of the line passing through the two points (-1, -2) and (3, -4), we can use the slope formula given by: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Plugging in the coordinates of the given points, we have: $$ m = \frac{-4 - (-2)}{3 - (-1)} $$
02

Simplify the Slope

Now, simplify the slope by performing the arithmetic operations in the formula: $$ m = \frac{-4+2}{3+1} $$ $$ m = \frac{-2}{4} $$ $$ m = -\frac{1}{2} $$ Now we have the slope (-1/2) of the line that passes through the points (-1, -2) and (3, -4).
03

Use the Point-Slope Form

Next, we will use one of the given points and the slope to find the equation of the line. Let's choose the point (-1, -2), and plug it into the point-slope form of a linear equation: $$ y - y_1 = m(x - x_1) $$ Substituting in the coordinates and slope, we get: $$ y - (-2) = -\frac{1}{2}(x - (-1)) $$
04

Simplify the Equation

Simplify the equation by performing the arithmetic operations and converting the equation into the slope-intercept form (y = mx + b). $$ y + 2 = -\frac{1}{2}(x + 1) $$ $$ y + 2 = -\frac{1}{2}x - \frac{1}{2} $$ Now subtract 2 from both sides: $$ y = -\frac{1}{2}x - \frac{1}{2} -2 $$ $$ y = -\frac{1}{2}x - \frac{5}{2} $$
05

Write the Final Equation

Now we have the equation of the line in slope-intercept form: $$ y = -\frac{1}{2}x - \frac{5}{2} $$ So, the equation of the line passing through the points (-1, -2) and (3, -4) is \(y = -\frac{1}{2}x - \frac{5}{2}\).

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Most popular questions from this chapter

Sketch the straight line defined by the linear equation by finding the \(x\) - and \(y\) -intercepts. $$ x+2 y-4=0 $$

Determine whether the points lie on a straight line. $$ A(-2,1), B(1,7), \text { and } C(4,13) $$

Consider the slope-intercept form of a straight line \(y=\) \(m x+b\). Describe the family of straight lines obtained by keeping a. the value of \(m\) fixed and allowing the value of \(b\) to vary. b. the value of \(b\) fixed and allowing the value of \(m\) to vary.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(p=m x+b\) is a linear demand curve, then it is generally true that \(m<0\).

Find an equation of the line that satisfies the given condition. The line passing through \((-5,-4)\) and parallel to the line passing through \((-3,2)\) and \((6,8)\)
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