Strategies, Sample Questions, and Random Ramblings.
July 18, 2015
GMAT Linear equations incorporate everything we discussed and that you practiced in the simplifying expressions section. In this chapter, we are going to discuss solving for single variables in an equation, also know as a linear equation.
Equation: A set of two different expressions that have the same value, denoted by an equal sign.
There are two types of equations that we will talk about: linear equations and quadratic equations. In a few sections, we will discuss quadratic equations. The main difference deals with exponents. A linear equation has no exponent value associated with any of the variables in the equation, where quadratic and other non-linear equations have an exponent on one or more of the variable terms (this includes exponents that are negative). When drawn on a coordinate plane, linear equations are in the shape of a straight line and non-linear equations have some type of curve.
The most basic rule to focus on when trying to solve equations is that what you do to one side of the equation you must do to the other. Any operation can be performed, as long as you do it to the other side as well.
Let’s look at this example:
4x = 3x + 7
Much like expressions, the main focus is going to be to combine like terms. In this instance, we want to put all of the x terms on one side of the equation. In order to do this we must subtract 3x from each side:
4x = 3x + 7
-3x =-3x + 7
Now that we have learned how to combine terms to solve for a variable, let’s look at a few ways the GMAT is going to challenge you by adding some wrinkles.
In this equation, you have a bunch of fractions. One thing to do would be to put the left hand side into common denominator format. In this case, 6 is the common denominator so you would perform the following calculations:
The first thing you might be wondering is how come we were able to do calculations on one side of the equation and not do them on the other side. In fact, however, we did not change the value of the left side as each fraction was simply multiplied by 1 (or a fraction that equals 1) so the value did not change. Next, we can do a step called “Cross-Multiplication.” First the result:
Then, how did we actually get there? Really, cross-multiplication is just a simplification of mathematical operations. All you are really doing is multiplying each side by the denominator to cancel it on one side and bring it to the other. Here is the first step:
Then, do the exact same to the other side:
You are left with the equation 28x = 6, which you know how to solve.
Going back to the original equation, there is a way to eliminate the fractions from the get go. To do this, simply multiply the entire equation by the LCM of all of the denominators. In the original case, this would be 12 and the operation would look like this:
While the two operations are the exact same, getting in the habit of eliminating fractions will be helpful for the more difficult problems.
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