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Chapter 2: Problem 34

Find a number \(t\) such that the line containing the points \((t, 2)\) and (3,5) is parallel to the line containing the points (-1,4) and (-3,-2) .

Short Answer

Expert verified
The value of \(t\) is 4, meaning that the line containing the points (4, 2) and (3, 5) is parallel to the line containing the points (-1, 4) and (-3, -2).

Step by step solution

01

Find the slope of the second line

Using the slope formula, the slope of the line through points (-1,4) and (-3, -2) can be found as follows: \[m = \frac{(-2 - 4)}{-3 - (-1)} = \frac{-6}{2} = -3\]
02

Write the slope expression for the first line

Now, using the coordinates of the first line (t, 2) and (3, 5), let's find the slope expression for the first line: \[m = \frac{(5-2)}{(3-t)} = \frac{3}{(3-t)}\]
03

Equate the slopes and solve for t

We need the lines to be parallel, which means their slopes must be equal. Therefore, we can equate the slopes and solve for t: \[-3 = \frac{3}{(3-t)}\] To solve for t, first multiply both sides by (3-t): \[-3(3-t) = 3\] Now simplify and solve for t: \[-9+3t = 3\] \[3t = 12\] \[t = 4\] So, the value of \(t\) is 4. This means the line containing the points (4, 2) and (3, 5) is parallel to the line containing the points (-1, 4) and (-3, -2).

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Most popular questions from this chapter

$$ \text { Suppose } p(x)=2 x^{6}+3 x^{5}+5 $$ (a) Show that if \(\frac{M}{N}\) is a zero of \(p\), then $$ 2 M^{6}+3 M^{5} N+5 N^{6}=0 $$ (b) Show that if \(M\) and \(N\) are integers with no common factors and \(\frac{M}{N}\) is a zero of \(p\), then \(5 / M\) and \(2 / N\) are integers. (c) Show that the only possible rational zeros of \(p\) $$ \text { are }-5,-1,-\frac{1}{2}, \text { and }-\frac{5}{2} \text { . } $$ (d) Show that no rational number is a zero of \(p\).

Find a polynomial \(p\) of degree 3 such that \(-2,-1,\) and 4 are zeros of \(p\) and \(p(1)=2\).

Show that $$ (a+b)^{3}=a^{3}+b^{3} $$ if and only if \(a=0\) or \(b=0\) or \(a=-b\).

Suppose \(p(x)=2 x^{4}+9 x^{3}+1\) (a) Show that if \(\frac{M}{N}\) is a zero of \(p\), then $$ 2 M^{4}+9 M^{3} N+N^{4}=0 $$ (b) Show that if \(M\) and \(N\) are integers with no common factors and \(\frac{M}{N}\) is a zero of \(p\), then \(M=-1\) or \(M=1\). (c) Show that if \(M\) and \(N\) are integers with no common factors and \(\frac{M}{N}\) is a zero of \(p\), then \(N=-2\) or \(N=2\) or \(N=-1\) or \(N=1\). (d) Show that \(-\frac{1}{2}\) is the only rational zero of \(p\).

Suppose \(s(x)=\frac{x^{2}+2}{2 x-1}\) (a) Show that the point (1,3) is on the graph of \(s\). (b) Show that the slope of a line containing (1,3) and a point on the graph of \(s\) very close to (1,3) is approximately -4 [Hint: Use the result of Exercise \(25 .]\)
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