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Rules about Combining Equations / Inequalities

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Rules about Combining Equations / Inequalities

by missrochelle » Sat Aug 21, 2010 2:09 pm
Hi -

I just wanted to know if anyone can simplify the rules for combining equations and how that differs from inequalities?

In which case does the coefficient have to be the same in order to combine. I read in MGMAT strategy guide problem:

a+b = 10, b+c = 12 and a+c = 16.

The explanation states that since the coefficients are 1, the best way to solve is via combination.
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by Stuart@KaplanGMAT » Sat Aug 21, 2010 8:41 pm
missrochelle wrote:Hi -

I just wanted to know if anyone can simplify the rules for combining equations and how that differs from inequalities?

In which case does the coefficient have to be the same in order to combine. I read in MGMAT strategy guide problem:

a+b = 10, b+c = 12 and a+c = 16.

The explanation states that since the coefficients are 1, the best way to solve is via combination.
Hi,

the rule for combining equations are much simpler than those for combining inequalities.

Here's the rule: you can always add or subtract equations.

Now, a more interesting question is: "when is it useful to combine equations?"

When you're solving systems of equations, and the coefficients of the variables you wish to eliminate are the same, combination works much more quickly than substitution.

For example, consider the question:

If x + 4y = 10 and x - 3y = -4, what's the value of y?

We certainly could solve this by substitution (i.e. isolate x in one equation, then substitute into the second), but combination is much quicker.

We note that x, the variable we want to eliminate, as the same coefficient in both equations. So, we line up the equations and subtract:

(x + 4y = 10)
-(x - 3y = -4)

and end up with:

7y = 14
y = 2

Note that if the coefficients don't line up already, we can make them line up by multiplying one or both equations by a constant. For example:

If 3x + 4y = 12 and x + 2y = 3, what's the value of y?

If we subtract the equations in their current form, we end up with 2x + 2y = 9, which isn't all that helpful. However, we can multiply both sides of the second equation by 3:

3(x + 2y) = 3(3)
3x + 6y = 9

Now if we subtract the new equation from the first, we get:

3x + 4y = 12
-(3x + 6y = 9)

-2y = 3
y = -(3/2)
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by missrochelle » Sun Aug 22, 2010 5:50 am
Thanks for your reply. this was a great explanation especially with the examples. My next question is --- when it comes to inequalities, can you also manipulate the way that you did in the second example? (Altering them so as to eliminate unneeded variables, and end up with the variable you are solving for?) .
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