Question 102508: Solve the system by addition or substitution.
–15x + 3y = 9
y = 3 + 5x
Please help me with this problem step by step if at all possible then maybe I can actually catch on, on how to do these problems. THANK YOU!!
Found 2 solutions by oberobic, bucky:
Answer by oberobic(2304)   (Show Source):
Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website! Given:
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–15x + 3y = 9
y = 3 + 5x
.
You are to solve this by addition or substitution.
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Let's try by substitution. The object of this method is to take one of the equations and solve
it for one of the variables in terms of the other variable. Then you substitute that into
the other equation and solve it. Sort of hard to explain, but easy to do.
.
Look at the bottom equation. It already shows that y equals 3 + 5x. So it is solved for
y in terms of x. Since we know that y = 3 + 5x, we can go to the top equation and replace y
by 3 + 5x. When we do that replacement, the top equation becomes:
.
-15x + 3(3 + 5x) = 9
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On the left side multiply the 3 times both of the terms in the parentheses and the equation
becomes:
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-15x + 9 + 15x = 9
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Wow! Notice that the -15x and the + 15x on the left side cancel each other out and we
are left with 9 = 9!!! What does that mean? From experience I know that this means that
there is not just one solution to this problem. This is a trick problem. Let's see why that is true.
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The top equation is:
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-15x + 3y = 9
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and the bottom equation is y = 3 + 5x
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Let's rearrange the bottom equation so the x and y terms are on the left side and the number
is on the right side. Do this by getting rid of the 5x on the right side and you can do that
by subtracting 5x from the right side. But if you subtract 5x from the right side you
must also subtract 5x from the left side to keep the equation in balance. When you do this
subtraction, the 5x on the right side disappears and a -5x appears on the left side. Therefore,
the bottom equation becomes:
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-5x + y = 3
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Now we can multiply both sides of this equation by 3 without changing its "balance"
and when we do that the bottom equation then becomes:
.
-15x + 3y = 9
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Notice anything unusual about that??? It's identical to the original top equation.
Therefore, it has the same graph as the top equation. This means that every coordinate
pair on the top equation is also a coordinate pair of the bottom equation. There is not just
one solution to this problem ... there are many, many solutions that will satisfy both
equations at the same time.
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Let's try an example. Suppose in the original bottom equation we let x equal 3. Then
the bottom equation becomes:
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y = 3 + 5(3) = 3 + 15 = 18
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Now let's go to the top equation and let x = 3. When we make that substitution into the
original top equation, we get:
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-15(3) + 3y = 9
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Multiplying out on the left side results in:
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-45 + 3y = 9
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Get rid of the -45 by adding 45 to both sides:
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3y = 54
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Solve for y by dividing both sides by 3 to get
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y = 54/3 = 18
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So the coordinate pair (3, 18) is a common solution for both problems. If you repeat this
same process for other values of x, you will continue to find that for each x you select and
substitute into both equations, the corresponding values for y in each of the equations is
the same. And whenever you put a common value of x into both equations, the corresponding
values of y from both equations will be equal. Many solutions ... an infinite number.
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You would have found out the same thing had you tried to solve the set of equations by
addition. Not just one unique solution, but many many solutions.
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Hope this helps you to understand this "trick" problem. Your problem does not have two
different equations. It just gives you the same equation twice ... in a little different
form, but nevertheless the same.
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