Question

50 POINTS

If a line passing through the points (a,1) and (9,8) is parallel to the line passing through the points (17,6) and (a+2,1) what is the value of A?

A = ????

Answer 1

Answer:

a = 30

Step-by-step explanation:

What are parallel lines?

• Parallel lines by definition are a pair of lines that extend forever and never intersect.

Important concepts and formulas

• Parallel lines have the same slope
• Slope = $\frac{y2-y1}{x_2-x_1}$ where the values of x and y derive from the given points

Creating an equation

Here, we are given that two lines that are parallel and we are given two coordinates that each line passes

Line 1 passes (a,1) and (9,8)

Line 2 passes (17,6) and (a+2,1)

Given that the two lines are parallel and that they pass these points, we are asked to find the value of a. We can do this by comparing the slopes of the two lines as we know they must be similar to be parallel. This means (slope of line 1) = (slope of line 2)

If slope = $\frac{y2-y1}{x_2-x_1}$

Then slope of line 1 = $\frac{8-1}{9-a}$

And slope of line 2 = $\frac{1-6}{(a+2)-17}$

If the slopes must be the same then,

$\frac{8-1}{9-a}=\frac{1-6}{(a+2)-17}$

Solving for a

We can now solve for a algebraically

$\frac{8-1}{9-a}=\frac{1-6}{(a+2)-17}$

==> simplify both sides

$\frac{7}{9-a}=\frac{-5}{(a+2)-17} }$

==> cross multiply

$-5(9-a)=7((a+2)-17)$

==> distribute

$-45 + 5a = 7a + 14 - 119$

==> combine like terms

$-45+5a=7a-105$

==> add 105 to both sides

$60+5a=7a$

==> subtract 5a from both sides

$60=2a$

==> divide both sides by 2

a = 30

Answer 2

Answer:

a = 30

Step-by-step explanation:

The slope of a line is a measure of its steepness and direction, and is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line:

$\boxed{\begin{array}{l}\underline{\textsf{Slope formula}}\\\\\large\text{ m=\dfrac{y_2-y_1}{x_2-x_1} }\\\\\textsf{where:}\\\phantom{w}\bullet\;\;m\; \textsf{is the slope.}\\\phantom{w}\bullet\;\;(x_1,y_1)\;\textsf{and}\;(x_2,y_2)\;\textsf{are two points on the line.}\end{array}}$

If a line passes through the points (a, 1) and (9, 8), then its slope is:

$m_1=\dfrac{8-1}{9-a}=\dfrac{7}{9-a}$

If a line passes through the points (17, 6) and (a+2, 1), then its slope is:

$m_2=\dfrac{1-6}{(a+2)-17}=\dfrac{-5}{a-15}$

If two lines are parallel, they have the same slope.

Therefore, to find the value of a, we can set the two slopes equal to each other, and solve for a:

$\begin{aligned}\dfrac{7}{9-a}&=\dfrac{-5}{a-15}\\\\7(a-15)&=-5(9-a)\\\\7a-105&=-45+5a\\\\2a&=60\\\\a&=30\end{aligned}$

Therefore, the value of a is:

$\huge\boxed{\boxed{a=30}}$