You can put this solution on YOUR website! Let x=one number
Then 14-x= the other number
Now we are told that
x-(14-x)=6 get rid of parens
x-14+x=6 add 14 to both sides
x-14+14+x=6+14 collect like terms
2x=20 divide both sides by 2
x=10 -------------------------------one of the numbers
14-x=14-10=4----------------------the other number
CK
10+4=14
14=14
and
10-4=6
6=6
Another way:
Let x=one number
And y= the other number
Now we are told that:
x+y=14-----------------------eq1
and
x-y=6----------------------------eq2
add eq1 and eq2
2x=20
-----and I bet that you can finish it.
You can put this solution on YOUR website! You have two unknown numbers. Represent them by X and Y.
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The problem tells you that the sum of these two numbers is 14. Represent this in equation
form as:
.
X + Y = 14
.
The problem also tells you that the difference of these two numbers is 6. Represent this
in equation form as:
.
X - Y = 6
.
There are several ways to solve this pair of equations. One way we can do it is by variable
elimination. Write the two equations together as follows:
.
X + Y = 14
X - Y = 6
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Notice that if we add the two equations vertically in columns we get X + X = 2X and for
the next column the +Y adds to -Y to cancel each other out. Finally notice on the other
side of the equal sign that the 14 adds to the 6 to give 20. So when we add the two equations
vertically we are left with the resulting equation:
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2X = 20
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We can now solve for X by dividing both sides by 2:
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2X/2 = 20/2
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And after the division we are left with:
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X = 10
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So we have one of the unknown numbers identified as 10. Since we know that the sum of the
two numbers must be 14, the missing 4 has to be the other number. This makes the two numbers
10 and 4. Notice that the difference in the two numbers is 10 - 4 and that equals 6
just as the problem said it should. This is just another check to ensure that our answer
is correct. The answers to the problem you posted are 10 and 4. Now for a lesson in using the
process of variable elimination to solve a pair of equations that have two unknowns.
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The reason that this method worked is that we can't solve a single equation for two unknowns.
If there are two independent equations for two unknowns, we should be able to solve them
if there is a single but common solution. Because of the values of the unknowns in the two
equations we originally wrote, we noticed that we could add the equations and one of the
variables (the Y) would cancel out so that we would be left with just one equation that
had one unknown. If the two equations did not have one of the variables equal but of
opposite sign to the same variable in the other equation, we would have to make it that way
in order to use variable elimination.
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For example only to show you how variable elimination works... suppose the two equations had been:
.
2X + 6Y = 20 and
3X + 4Y = 10
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We could multiply the entire top equation (all terms) by 3 to convert it to:
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6X + 18Y = 60
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and we could multiply both sides of the bottom equation (all terms) by -2 to make it
become:
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-6X - 8Y = -20
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So we have converted the original pair of equations to:
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+6X + 18Y = 60 and
-6X - 8Y = -20
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We have made the terms containing X equal but opposite in sign. Notice that if we now
add the two equations vertically in columns that the +6X and the -6X will cancel each
other and the rest of the sums will leave us +18Y - 8Y = 10Y and on the other side we
have +60 -20 = +40. So we have reduced the pair of equations to:
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10Y = 40
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And dividing both sides of that equation by 10 gives us Y = 4
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Now that we know Y is equal to 4, we can return to either of the original equations
in this example and substitute 4 for Y and solve for X. One of the original two equations
we started with was:
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3X + 4Y = 10
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Substituting 4 for Y results in:
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3X + 4(4) = 10
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And multiplying the 4 times 4 converts this to:
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3X + 16 = 10
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Getting rid of the +16 on the left side by subtracting 16 from both sides we are left with:
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3X = -6
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and we can solve for X by dividing both sides by 3 to get X = -2. So the answers to this sample
problem are X = -2 and Y = +4.
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Hope this helps you to understand the problem you posted and gives you some insight into
solving two equations with two unknowns by using the process of eliminating a variable
from the pair of equations.
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