Question 105710: Here is the question:
Solve the following system of equations using elimination. Show all of your work. Once you solve the system, explain each step in solving the process. Include what you did and why you did it. State the solution and explain what the solution actually tells you about the system of equations.
6x+3y=27
-4x+7y=27
This is what I started, but I'm not sure if it's right....
7(6x+3y=27) = 42x+21y=189
-3(-4x+7y=27) = 12x-21y=81
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Given:
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+6x+3y=27
-4x+7y=27
This is what I started, but I'm not sure if it's right....
You have the right idea for solving this equation by eliminating one of the variables.
To use this method you need to get one of the variables in one equation to equal the same
variable term in the other equation. You can do this by multiplying both equations by an
appropriate number. You have chosen to eliminate the variable y terms. And you are multiplying
the top equation by +7 and the bottom equation by -3. This work is shown below:
+7(+6x+3y=27) = 42x+21y = 189 <== OK
-3(-4x+7y=27) = 12x-21y = -81 <== note you had +81 and I changed it to -81 because -3*27 = -81
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You now have two equations:
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42x + 21y = 189 and
12x - 21y = -81
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Now you can add these two equations vertically. (You are adding equals to equals, so
the resulting equation is still balanced.) When you add vertically, 42x + 12x equals 54x.
And next, +21y added to -21y results in a cancellation ... so the y terms disappear.
Finally, on the other side, 189 added to -81 is 108. So the vertically adding of the two
equations results in:
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54x = 108
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Solve for x by dividing both sides by 54 to get:
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x = 108/54 = 2
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You now know that x is 2. You can return to either of the two equations that you were given
originally and in the one you select you can substitute 2 for x and then solve for y.
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Let's return to the original equation 6x + 3y = 27. If we substitute 2 for x the term 6x
becomes 6*2 = 12. So the equation becomes:
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12 + 3y = 27
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Get rid of the 12 on the left side by subtracting 12 from both sides to reduce the equation
to:
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3y = 15
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Finally solve for y by dividing both sides of this equation by 3 to get:
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y = 15/3 = 5
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In summary, the answer to this problem is x = 2 and y = 5.
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What does this tell you? It tells you that the (x, y) point [which is (2, 5)] satisfies
both equations. That is, if you go to the two equations you were given and let x equal 2 and
y equal 5, both sides of each of the two equations are equal. It also tells you that if you
graph both equations, the two graphs will cross at the point (2, 5).
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Hope this helps you with understanding the problem and how to get the answer. You started
out on the right track.
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