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Chapter 2: Problem 29

Find a number \(t\) such that the point \(\left(t, \frac{t}{2}\right)\) is on the line containing the points (2,-4) and (-3,-11) .

Short Answer

Expert verified
The value of \(t\) such that the point \(\left(t, \frac{t}{2}\right)\) is on the line containing the points (2,-4) and (-3,-11) is \(\boxed{\frac{68}{9}}\).

Step by step solution

01

Find the equation of the line containing the points (2,-4) and (-3,-11)

Given two points, we can find the equation of the line using the slope-point form, which is given by: \[y - y_1 = m(x - x_1)\] where \(m\) is the slope of the line, and \((x_1, y_1)\) is one of the given points. To find the slope, we can use the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\] Take the points (2,-4) and (-3,-11) as \((x_1, y_1)\) and \((x_2, y_2)\), respectively. Next, find the slope: \[m = \frac{-11 - (-4)}{-3 - 2} = \frac{-7}{-5} = \frac{7}{5}\] Now, using the slope-point form, plug in the slope and one of the points (let's use the point (2,-4)) to find the equation of the line: \[y - (-4) = \frac{7}{5}(x - 2)\]
02

Substitute the given point and solve for t

We need to find the value of \(t\) such that the point \(\left(t, \frac{t}{2}\right)\) lies on the line found in Step 1. Substitute the point into the equation: \[\frac{t}{2} + 4 = \frac{7}{5}(t - 2)\] Now, solve for \(t\): \[\frac{t}{2} + 4 = \frac{7t}{5} - \frac{14}{5}\] First, multiply both sides by 10 to eliminate the fractions: \[5t + 40 = 14t - 28\] Next, move all the t terms to one side and constant terms to the other side: \[14t - 5t = 40 + 28\] \[9t = 68\] Now, divide by 9: \[t = \frac{68}{9}\] So, the value of \(t\) such that the point \(\left(t, \frac{t}{2}\right)\) is on the line containing the points (2,-4) and (-3,-11) is \(\boxed{\frac{68}{9}}\).

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