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Chapter 2: Problem 26

Find a number \(c\) such that the point \((c,-19)\) is on the line containing the points (2,1) and (4,9) .

Short Answer

Expert verified
The value of \(c\) is -3, making the point \((-3,-19)\) on the line containing the points \((2,1)\) and \((4,9)\), with the line's equation being \(y = 4x - 7\).

Step by step solution

01

Find the slope of the line

First, we need to find the slope of the line containing the points (2,1) and (4,9). To do this, we use the formula: \(slope = \frac{y2 - y1}{x2 - x1}\) where (x1, y1) is the first point (2, 1) and (x2, y2) is the second point (4, 9). Plugging the values into the formula, we get: \(slope = \frac{9 - 1}{4 - 2} = \frac{8}{2} = 4\) The slope of the line is 4.
02

Find the equation of the line

Now that we have the slope and a point on the line, we can use the point-slope form of the equation of a line to find the equation of the line. The point-slope form is given by: \(y - y1 = slope \times (x - x1)\) Using the slope we found in step 1 and one of the given points, let's use (2,1), we get: \(y - 1 = 4 \times (x - 2)\) Now we can rewrite this equation in the slope-intercept form (y = mx + b): \(y = 4(x - 2) + 1\) \(y = 4x - 8 + 1\) \(y = 4x - 7\) The equation of the line is \(y = 4x - 7\).
03

Substitute the point and solve for c

Now that we have the equation of the line, we can substitute the coordinates of the point \((c, -19)\) into the equation to find the value of \(c\). Plug the values of the point \((c, -19)\) into the equation: \(-19 = 4c - 7\) Now we just need to solve for \(c\): Add 7 to both sides of the equation: \(-12 = 4c\) Divide both sides by 4: \(-3 = c\) So the value of \(c\) is -3, and the point \((-3,-19)\) is on the line containing the points \((2,1)\) and \((4,9)\).

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Most popular questions from this chapter

Explain why the polynomial \(p\) defined by $$ p(x)=x^{6}+100 x^{2}+5 $$ has no real zeros.

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

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 \(x\).

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