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Chapter 2: Problem 25

Find a number \(c\) such that the point \((c, 13)\) is on the line containing the points (-4,-17) and (6,33) .

Short Answer

Expert verified
The value of $c$ is 2, so the point (2, 13) lies on the line containing the points (-4, -17) and (6, 33).

Step by step solution

01

Find the slope of the line through (-4, -17) and (6, 33).

We begin by finding the slope of the line containing the two points (-4, -17) and (6, 33). The slope (m) is calculated using the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) using the coordinates of the two points. Here, \((x_1, y_1) = (-4, -17)\) and \((x_2, y_2) = (6, 33)\). \[m = \frac{33 - (-17)}{6 - (-4)}\]
02

Simplify the slope.

Simplify the expression we got in step 1 to find the slope. \[m = \frac{33 + 17}{6 + 4}\] \[m = \frac{50}{10}\] \[m = 5\] Now we know the slope of the line is 5.
03

Set up an equation to find c.

Now, we will use the point-slope form of a linear equation to find the value of c which leads to the point (c, 13) lying on the same line. The point-slope form of a linear equation is given by: \[y - y_1 = m(x - x_1)\] We will plug in the point (-4, -17) (x1, y1) along with the slope m = 5, as well as the point (c, 13) into the equation. \[13 - (-17) = 5(c - (-4))\]
04

Solve the equation for c.

Now we will solve the equation from step 3 for c. \[13 + 17 = 5(c + 4)\] \[30 = 5(c + 4)\] \[6 = c + 4\] Now, subtract 4 from both sides of the equation: \[6 - 4 = c\]
05

Find the value of c.

Now we have the equation: \[2 = c\] So the value of c is 2. Therefore, the point (2, 13) lies on the line containing the points (-4, -17) and (6, 33).

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Most popular questions from this chapter

Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{4 x-5}{x+7} $$

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

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

A new snack shop on campus finds that the number of students following it on Twitter at the end of each of its first five weeks in business is 23,89,223 , \(419,\) and \(647 .\) A clever employee discovers that the number of students following the new snack shop on Twitter after \(w\) weeks is \(p(w),\) where \(p\) is defined by $$p(w)=7+3 w+5 w^{2}+9 w^{3}-w^{4}$$ Indeed, with \(p\) defined as above, we have \(p(1)=23,\) \(p(2)=89, p(3)=223, p(4)=419,\) and \(p(5)=647\) Explain why the polynomial \(p\) defined above cannot give accurate predictions for the number of followers on Twitter for weeks far into the future.
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