Today, we’re covering the basics for GED algebra prep. Algebra makes up a large portion of the GED Mathematical Reasoning exam, so you’ll want to make sure you have a handle on it going in to test day.
Algebra deals with formulas and unknown values. An unknown value is represented by a variable. Often you will see the variables x and y, but any letter or symbol can be a variable.
Solving Basic Equations (1 variable)
Solving an algebraic equation means to find the values of the variables. The simplest types of equations to solve are those with only one variable.
To solve an equation with one variable, your goal is to isolate the variable. This means to get the variable alone on one side of the equation. How do you do this? By using inverse operations to undo what is being done in the equation.
You might think of inverse operations as being opposites. Each inverse operation undoes whatever its partner does. There are two sets of inverse operations:
- addition and subtraction
- multiplication and division
Solving One-Step Equations
Let’s see how these inverse operations work to isolate the variable in a very basic algebra equation:
This equation is saying that some unknown amount (x), minus 4, is equal to 0. To find the value of x, you need to isolate it, or get it on one side of the equation by itself. Right now it is on the same side as minus 4. To get rid of that minus 4, you need to add 4. But:
- IMPORTANT! Whatever you do to one side of an equation, you must also do to the other side.
If you add 4 to the left side of the equation, you also need to add 4 to the right side of the equation:
Here’s what adding 4 does to both sides of the equation:
- On the left side, if you add 4 to a negative 4, you get 0, which is nothing, which means you have met your goal of isolating the variable.
- On the right side, if you add 4 to 0, you get 4.
So, you end up with:
And the equation is solved.
Solving Two-Step Equations
A two-step equation still only has one variable, but it involves two steps to isolate the variable. Here’s an example:
Isolating x in this equation involves two steps:
- Eliminating the plus 2 on the left side.
- Eliminating the coefficient 3 on the left side. (A coefficient is the number a variable is being multiplied by.)
To eliminate the plus 2, you need to subtract 2 from both sides of the equation. Here’s what that looks like:
To eliminate the coefficient 3, you need to divide each side of the equation by 3. Here’s what that looks like:
You’ve now isolated the variable, and so the equation is solved.
Solving Basic Inequalities (1 variable)
Inequalities are used to compare quantities. There are 4 basic inequality signs:
|≥||Greater than or equal to|
|≤||Less than or equal to|
Inequalities are solved similarly to equations, with one very important exception:
- IMPORTANT! Whenever you multiply or divide by a negative number, you must flip the inequality sign.
Otherwise, complete the same steps to solve an inequality as you would an equation.
Here’s an example:
Solving this inequality has three steps:
- Eliminating the 2 on the left side
- Eliminating the -7 on the left side
- Flipping the inequality sign
To eliminate the 2, you need to subtract 2 from both sides of the equation:
To eliminate the -7, you need to divide both sides of the equation by -7:
Since you divided by a negative number in the previous step, you need to flip the inequality sign:
Solving System of Equations (2 variables)
A system of equations is a set of related equations that you work simultaneously to solve. The equations will contain two variables, for example, x and y. Since you can’t find two unknown variables in one equation, a system of equations is such that you need both equations to find both variables.
Solving a system of equations with two variables involves three steps:
- Isolate x in the first equation, finding an expression that describes it.
- Substitute the expression of x into the second equation, solving for y.
- Substitute the value of y into either equation, finding the value of x.
Here’s an example:
To isolate x in the first equation, divide both sides by 4 (remember to divide EVERY term by 4):
Now, substitute the expression describing x into the second equations:
To solve for y, first use the distributive property to remove the parentheses:
Then, subtract 9 from both sides:
Next, subtract 3y from both sides:
Finally, divide both sides by -1:
Now that you know the value of y, you can substitute this value into either equation and solve for x:
Now you know the value of both variables, and the system of equations is solved.
Working with Polynomials
Polynomials are expressions with more than one term. A term can be a variable, number, or exponent. You can add, subtract, multiply, and divide polynomials. You will also have occasions when you will need to factor polynomials.
Follow two steps to add polynomials:
- Rearrange the polynomials by like terms.(Like terms are terms that have the same variable and exponent.)
- Combine like terms by adding them.
When adding (and also when subtracting, multiplying, or dividing) polynomials, the goal is not to find the value of variables, but to combine the two polynomials into one simplified polynomial.
Rearrange the two polynomials into one polynomial, grouping like terms together:
Simplify the expression by combining like terms:
Subtracting polynomials is similar to the adding process, except for one important difference in the first step:
- Reverse the signs in the polynomial you are subtracting. This means turning positive signs into negative signs, and vice versa.
- Rearrange the polynomials by like terms.
- Combine like terms by adding them.
For example, to subtract these polynomials, follow the steps below:
8x^2 -14x +7
To multiply polynomials, you need to multiply each term of the first polynomial by each term of the second polynomial. One most common type of polynomial multiplication you’ll see on the GED is the multiplication of binomials, which you do using these 5 steps:
- Multiply the first terms of each binomial.
- Multiply the outer terms of each binomial. This means the first term in the first binomial, and the second term in the second binomial.
- Multiply the inner terms of each binomial. This means the second term in the first binomial, and the first term in the second binomial.
- Multiply the last terms of each binomial.
- Combine like terms.
ALGEBRA TIP: Use the acronym FOIL (first, outer, inner, last) to help you remember the order of multiplication when multiplying binomials.
Multiply the first terms of each binomial:
Multiply the outer terms of each binomial:
Multiply the inner terms of each binomial:
Multiply the last terms of each binomial:
Combine like terms to simplify the expression.
Dividing polynomials usually involves two steps:
- Split the numerator. This will result in multiple fractions instead of one. Each fraction will have the same denominator. Each numerator will be one of the terms in the numerator.
- Simplify each fraction.
Split the numerator. Each term will become a separate numerator. Each fraction will have a denominator of 5x:
Simplify the first fraction:
Then, simplify the second fraction.
There are numerous methods for factoring the numerous types of polynomials. The most basic method, however, is to find the greatest common factor (GCF) of all the terms in the polynomial.
- IMPORTANT! The GCF must be common to each term in the polynomial, including terms without variables. For example, 4x is not a factor of 4. On the other hand, 4 IS a factor of 4x.
Factoring a polynomial in this way has three steps:
- Identify the GCF for ALL terms.
- Pull out the GCF.
- Put the remaining factors in parentheses.
The greatest common factor is 4x, since 4x divides evenly into each term.
To set up the factorization, pull out the GCF, and set it outside a set of parentheses:
Go through the polynomial term-by-term, determining which factor remains after pulling out 4x. In other words, ask yourself, what do I have to multiply by 4x to get the original term in the polynomial?
For the first term:(4x)(3x2)=12x3
For the second term:(4x)(-2x)=-8x2
For the third term:(4x)(1)=4x
Put these three terms into the parentheses with the GCF out front: 4x(3x2-2x+1)
Learn more about what’s on the GED math test.
Put your algebra skills to the test with some math practice questions.