Question 556703: Hi please help me I'm trying to study and I don't get this:
Solve each system of equations
2a=8
a+b=2
thank you
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Given to solve for a & b:
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2a = 8
a + b = 2
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There are a couple of ways you can do this problem.
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One way is by substitution. Because the first equation has only one variable, the unknown a, it can be solved for a. Once you do that you can replace the a in the second equation and solve for b. Let's do it:
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Start with the first (or top) equation:
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2a = 8
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Divide both sides by 2 because a is multiplied by 2. When you do this division of both sides the result is:
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a = 4
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So now you know the value of a. You can go to the second equation and replace the a with 4. When you do that replacement, the second equation becomes:
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4 + b = 2
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Get rid of the 4 on the left side by subtracting 4 from both sides. When you do that the equation becomes:
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4 - 4 + b = 2 - 4
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On the left side the 4 - 4 becomes zero (or just disappears) and on the right side the 2 - 4 equals -2. What you then have is:
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b = -2
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So you now know that the answers are a = 4 and b = -2.
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Another way to solve this pair of equations is by variable elimination. Here's the way you can do that.
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First recognize that the top equation can be written as:
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2a + 0b = 8
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The term zero times b is just a "place holder" that makes the top equation look like the second equation. Zero times b is just zero, meaning that there is really nothing there.
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Now you can write the two original equations as:
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2a + 0b = 8 and
a + b = 2
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Now what you want to do is make the term in one of these equations identical to the same term in the other. We could multiply the top equation (all terms on both sides) by 1/2 to make the top equation become a + 0b = 4 and then we would have the pair of equations:
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a + 0b = 4
a + b = 2
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or we could have multiplied the bottom equation (all terms on both sides) by 2 to make the pair of equations:
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2a + 0b = 8 and
2a + 2b = 4
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Now we can subtract vertically in columns --- subtracting the bottom equation from the top equation. (I'll work the first set of equations in which we multiplied the top equation by 1/2. For practice, you can work the second set of equations where we multiplied the bottom equation by 2. You should get the same answers if you follow the general procedure.) Notice that in the first column the a on top minus the a below it goes to zero and disappears. In the second column the 0b subtracting the b below it results in -b. And on the right side the 4 on top minus the 2 below it results in +2. So after these subtractions we are left with:
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-b = +2
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and to solve for +b (or just plain b) we multiply both sides by -1 to change this equation to:
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b = -2
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This is the same answer that we got by using substitution earlier. Then all you need to do is go to either of the two original equations and substitute -2 for b. Then solve for a. Let's go to the original second equation:
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a + b = 2
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Substitute -2 for b and this equation becomes:
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a - 2 = 2
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Get rid of the -2 on the left side by adding 2 to both sides and you have:
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a - 2 + 2 = 2 + 2
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and this simplifies to:
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a = 4
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This is the same answer that we got before.
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I hope this helps clarify how you might do problems of this type. Most problems of this type will involve both variables in the two equations. Therefore, you will probably want to use variable elimination which is the second method we used above. This will involve multiplying one or both of the equations by a constant (or constants) so that the equations have one term in common that can be eliminated by subtracting the two equations in vertical columns. When you do that, you will have just a term with one variable on the left side and just a constant on the other side of the equal sign.
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You'll need to do some thinking about this, but after a few more sample problems of this nature, you'll get the idea of just how this works. Good luck to you ... and keep working at it.
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