Question 524185: How do you find out the values of a, b and c if a+b=12 and a+c=13 and b+c=10?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Given:
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a + b = 12
a + c = 13
b + c = 10
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First, note that there are 3 variables, namely a, b, and c. Therefore, if there is a solution, we need 3 independent equations, and we do have that. (If there were only 2 equations, it would be a problem.)
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What we need to do is to pick one of the equations and use the other two equations to get the one we chose so that it has only one variable in it. Then we can solve for that one variable. It sounds harder than it really is.
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We can choose any of these three equations, but let's take the first one:
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a + b = 12
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Let's replace the b in it. The only other equation that we have with b in it is the last equation:
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b + c = 10
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Solve this equation for b by subtracting c from both sides to get:
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b = 10 - c
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Now we can go back to the first equation and substitute 10 - c in place of the b. When we do that the first equation becomes:
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a + 10 - c = 12
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Just for reference let's call this equation 4. Now let's get rid of the c in equation 4. To do this we use the second equation:
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a + c = 13
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Solve for c by subtracting a from both sides to get:
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c = 13 - a
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Return to equation 4 and substitute 13 - a for c to get:
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a + 10 - (13 - a) = 12
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Notice that we now have an equation that has only one variable, and it is a. An equation with only one variable can be solved for that particular variable. Let's proceed.
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Since the parentheses are preceded by a minus sign, when we remove the parentheses we need to change the signs of the terms within them. Doing that results is the equation becoming:
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a + 10 - 13 + a = 12
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On the left side combine the +10 and the -13 to get -3 and then we have:
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a - 3 + a = 12
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Get rid of the - 3 on the left side by adding a + 3 to both sides. When you do that the equation becomes:
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a + a = 15
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Add the two a's to get:
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2a = 15
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and solve for a by dividing both sides by 2 to get:
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a = 15/2 = 7.5
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Now we can return to the original first equation that we were given:
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a + b = 12
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Substitute 7.5 for a:
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7.5 + b = 12
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And subtract 7.5 from both sides to find that:
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b = 12 - 7.5 = 4.5
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We also can use the original second equation that we were given to solve for c:
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a + c = 13
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Substitute 7.5 for a and we have:
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7.5 + c = 13
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Subtract 7.5 from both sides and the equation becomes:
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c = 13 - 7.5 = 5.5
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We now have our answers of a = 7.5, b = 4.5, and c = 5.5. As a check we can go to the original last equation to see if:
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b + c = 10
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Substitute 4.5 for b and 5.5 for c and this last equation becomes:
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4.5 + 5.5 = 10
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which when you add the two terms on the left side says:
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10 = 10
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This is obviously true, and gives us confidence that our answers for a, b, and c are correct.
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In summary, one way of solving three independent equations with three unknowns can be to select one equation, and eliminate variables by using the other two equations to do so. The goal is to end up with one equation having only one variable which can be solved for that variable. Then, using the solution for that one variable, work your way back through the other equations to determine the other variables.
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Hope this helps you to understand the problem a little better.
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