Question 81496: Solve each of the following systems by substitution.
8x - 4y = 16
y = 2x - 4
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Solve the following system by substitution:
.
8x - 4y = 16
y = 2x - 4
.
Since the bottom equation is already solved for y in terms of x, let's substitute the right
side of the bottom equation for y in the top equation. With that substitution the top equation
becomes:
.
8x - 4(2x - 4) = 16
.
Do the distributed multiplication on the left side to get:
.
8x - 8x + 16 = 16
.
Notice that the two terms that contain x cancel each other. That reduces the equation
to just:
.
+16 = +16
.
What does this mean? It means that the two equations are always satisfied no matter what
value of x you choose. For that to be the case, both equations must be the same so that
every solution of one of them is also a solution of the other.
.
Another way to look at this is to say to yourself, let me solve the top equation for y in
terms of x. Start with the top equation:
.
8x - 4y = 16
.
Divide all the terms on both sides by the common factor of 4. When you do that division,
the top equation becomes:
.
2x - y = 4
.
Next subtract 2x from both sides and you get:
.
-y = -2x + 4
.
Finally, to solve for y, multiply all the terms on both sides by -1 and you get:
.
y = +2x - 4
.
Look at that! It turns out that the top equation is the same as the bottom equation.
So every solution that works in the top equation must also work in the bottom equation.
This also means that the graph of the top equation lies on top of the graph of the bottom
equation. Therefore, there are an infinite number of common solutions.
.
Hope this helps you to understand how the problem has misled you by asking you to solve
using substitution, only for you to find out that all the variables in the equation after
substitutions are made cancel out or disappear.
.
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