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# finding the value of a when two lines have the same slope

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a graph of 6x - ay = 8 and 2x + 3y = 12 shows two lines that have the same slope. what is the value of a?

Aug 1, 2017

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Not all authorities agree on the standard form of a line; some believe it to be $$Ax+By=C$$while others believe it to be $$y=mx+b$$. Regardless of which form you use as your standard, the answer will not be different. For this problem, I will use the form $$Ax+By=C$$ because the equations are already in that form. I will, therefore, use 2 different method to solve this problem because of the likelihood of ambiguity.

Method 1

For a system, however, we have this form:

$$A_1x+B_1y=C_1$$

{ $$A_2x+B_2y=C_2$$

Before I start solving, I need to straighten out a rule

1. If $$\frac{A_1}{A_2}=\frac{B_1}{B_2}\neq\frac{C_1}{C_2}$$, then the system has no solution.

2. If $$\frac{A_1}{A_2}=\frac{B_1}{B_2}=\frac{C_1}{C_2}$$, then the system has infinitely many solutions.

Why is this knowledge beneficial? Well, if a system has no solution or has infinitely many of them, then that means that the lines of each equation must be parallel. And by definition, if two lines are parallel, they must have the same slope. Therefore, let's look at your system:

{$$6x-ay=8$$

{$$2x+3y=12$$

Let's see what happens when we try to put the numbers over one another:

Does $$\frac{6}{2}=\frac{-a}{3}\neq\frac{8}{12}$$? Yes, 6/2=3, and that is definitely not equivalent to 8/12, so we know that we have the first case. To solve for a, though, we must set it equal to $$\frac{A_1}{A_2}$$, which is equal to 3. Let's do that:

 $$3=\frac{-a}{3}$$ Multiply by 3 on both sides. $$9=-a$$ Divide by -1. $$a=-9$$

What if you want to check your answer? To check your answer, you can graph it and see if the lines ever intersect. If they do not intersect, then you have done the problem correctly.

You can see the graph for this method here at https://www.desmos.com/calculator/9nimq6ig4q

Method 2

For method 2, I will convert both equations to y=mx+b form and then go from there:

First, I will convert 6x - ay = 8:

 $$6x-ay=8$$ Subtract 6x on both sides. $$-ay=-6x+8$$ Divide by -a on both sides of the equaton. $$y=\frac{-6x+8}{-a}$$ I will break the fraction using the rule that $$\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$$ $$y=\frac{-6x}{-a}+\frac{8}{-a}$$ $$y=\frac{6}{a}x-\frac{8}{a}$$

Now, I will do the same to the other equation:

 $$2x+3y=12$$ Our objective is to solve for y. To accomplish this, subtract x from both sides. $$3y=-2x+12$$ Divide by 3 on both sides. $$y=\frac{-2x+12}{3}$$ Break up the fraction using the same process as above. $$y=-\frac{2}{3}x+4$$

Of course, the generalized equation for a line is y=mx+b where m is the slope. In order for 2 lines to have the same slope, we must set them equal to one another and solve for a:

 $$-\frac{2}{3}=\frac{6}{a}$$ Multiply by a on both sides of the equation. $$\frac{a}{1}*\frac{-2}{3}=\frac{6}{a}*\frac{a}{1}$$ Simplify both sides of the equation. $$\frac{-2a}{3}=6$$ Multiply by 3 on both sides. $$-2a=18$$ Divide by -2 on both sides. $$a=-9$$ Just like with the method above, we got the same answer

You can see the graph for this method here at https://www.desmos.com/calculator/rmv645u9ij

Aug 1, 2017