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Chapter 1: Problem 18

If the line passing through the points \((a, 1)\) and \((5,8)\) is parallel to the line passing through the points \((4,9)\) and \((a+2,1)\), what is the value of \(a\) ?

Short Answer

Expert verified
The value of \(a\) is \(\boxed{26}\).

Step by step solution

01

Calculate the slope of the first line

To find the slope of the line that passes through the points \((a, 1)\) and \((5,8)\), we can use the formula: \( \frac{y_2 - y_1}{x_2 - x_1}\) where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. Plugging in the coordinates, we get: \(m_1 = \frac{8 - 1}{5 - a} = \frac{7}{5 - a}\)
02

Calculate the slope of the second line

Now, let's find the slope of the line passing through the points \((4, 9)\) and \((a + 2, 1)\). Using the same formula as above, we have: \(m_2 = \frac{1 - 9}{(a + 2) - 4} = \frac{-8}{a - 2}\)
03

Set the slopes equal and solve for \(a\)

Since the lines are parallel, their slopes are equal. Therefore, we can write the following equation: \(m_1 = m_2\). Substituting the expressions we found for the slopes, we get: \(\frac{7}{5 - a} = \frac{-8}{a - 2}\) To solve this equation for \(a\), we can cross-multiply, obtaining: \(-8(5 - a) = 7(a - 2)\) Expanding the equation, we arrive at: \(-40 + 8a = 7a - 14\)
04

Solve for \(a\)

Now, let's isolate \(a\) on one side of the equation: \(8a - 7a = -14 + 40\) \(a = 26\) So, the value of \(a\) is \(\boxed{26}\).

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