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Chapter 2: Problem 14

Find the equation of the line that contains the points (-3,2) and (-5,7)

Short Answer

Expert verified
The equation of the line containing the points (-3, 2) and (-5, 7) is \(y = -\frac{5}{2}x -\frac{11}{2}\).

Step by step solution

01

Find the slope using the coordinates of two points.

The slope of a line with points (x1, y1) and (x2, y2) is given by the formula \(\frac{y2 - y1}{x2 - x1}\). Given the points (-3, 2) and (-5, 7), we can substitute the values into this formula to calculate the slope (m): m = \(\frac{7 - 2}{-5 - (-3)} = \frac{5}{-2}\) So, the slope of the line is -\(\frac{5}{2}\).
02

Choose a point to use with the point-slope form.

We can use either point (-3, 2) or point (-5, 7) along with the slope for the point-slope form of a linear equation. Let's use the point (-3, 2). The point-slope form of a linear equation is given by: \(y - y1 = m(x - x1)\)
03

Substitute the point and slope values into the point-slope form equation.

Replacing the variables with the values from step 1 (slope) and step 2 (point), we get: \(y - 2 = -\frac{5}{2}(x - (-3))\)
04

Convert the equation to slope-intercept form.

In order to simplify the equation, we first need to distribute the slope -\(\frac{5}{2}\) through the parenthesis: \(y - 2 = -\frac{5}{2}x -\frac{15}{2}\) Now, we add 2 (or, equivalently, \(\frac{4}{2}\)) to both sides of the equation to rewrite it in slope-intercept form: \(y = -\frac{5}{2}x -\frac{11}{2}\) And that is our resulting line equation in slope-intercept form, which contains both given points (-3, 2) and (-5, 7).

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Most popular questions from this chapter

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r(x))^{2} $$

Suppose \(M\) and \(N\) are odd integers. Explain why $$ x^{2}+M x+N $$ has no integer zeros.

Explain why the composition of a polynomial and a rational function (in either order) is a rational function.

Let \(p\) be the polynomial defined by $$p(x)=x^{6}-87 x^{4}-92 x+2$$. (a)Use a computer or calculator to sketch a graph of \(p\) on the interval [-5,5] . (b) Is \(p(x)\) positive or negative for \(x\) near \(\infty ?\) (c) Is \(p(x)\) positive or negative for \(x\) near \(-\infty ?\) (d) Explain why the graph from part (a) does not accurately show the behavior of \(p(x)\) for large values of \(x\).

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r(x))^{2} t(x) $$
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