ACT Math: Elementary Algebra

Algebra is a branch of mathematics that describes equations and arithmetic “symbolically.” Whereas in normal arithmetic an expression like “2+2” has a definite answer, in algebra unidentified numbers can be symbolically represented by variables; “a,b,c,x,y, or z”. A simple addition in algebra could look like this “a+b=c”. With these concepts, we can look at an algebraic equation.
“x+6 = 16”
When we are presented with an algebraic equation like this, we can actually “solve the equation” which just means we can find the numeric value of the ‘x‘ variable in the equation. The numeric value of x that satisfies the equation, keeps the equation true, is called the solution to the equation. You’ll see lots of algebra questions on the ACT Math test.
 

So how do we start solving any equation? Well, there are a number of operations we can do on any equation and these operations are our main tools in Algebra. In Algebra, the number one concept to understand is that on equality. Any equation, or equality, can be modified by doing the same arithmetical operation to BOTH SIDES of the equation. For example, let’s take another look at the equation “x+6 = 16”. Let’s subtract a constant from both sides of the equation, in this case 6.
(x+6) – 6 = (16) – 6 On the left side of the equation, 6-6 = 0, so those two terms cancel out leaving only x. And on the right side of the equation, 16-6 = 10, so 10 remains on the right side. So after we have subtracted 6 from both sides, we are left with this equation: x = 10. Notice the form of this final equation. What is left is actually the solution to the equation! If we now know that x is equal to 6, then if we substitute 6 for x in the original equation, we’ll find that the equality is still true.
X + 6 = 16 : (10) + 6 = 16 Notice that 10 + 6 does equal 16, so we know that x = 10 is a correct solution to the equation. This was a very simple example of solving an algebraic equation. Once again, the most important concept to understand about solving algebraic equations is how to do operations on BOTH sides of the equation. The main method of solving a simple equation like this is called “isolating the variable” where on one side of the equation is only the variable term and on the other side is everything else. With this, we have the basics of elementary algebra.

Let’s try a harder problem and go through all the concepts so far: 3x + 24 = 8 + 5x
Notice that there are 4 terms in the equation; 3x and 24 in the left expression, 8 and 5x in the right expression. So if we want to solve the equation, we need to isolate the variable, ‘x‘. In order to do that, we have to do arithmetical operations to each side of the equation. Since there a term with x on both sides of equation, we can do an operation to manipulate the equation so that x is only on one side of the equation. Since 3x is less than 5x, we can subtract 3x from both sides of the equation thereby leaving the equation with x on only one side of the equation.
(3x + 24) – 3x = (8 + 5x) – 3x :
3x – 3x + 24 = 8 + 5x – 3x :
24 = 8 + 2x
So now that we have only one x term on one side of the equation, we can take the next step in isolating x by subtracting 8 from both sides of the equation.
(24) – 8 = (8 + 2x) – 8 :
16 = 2x
One more step and we can completely isolate x thereby finding the solution for the equation. Let’s divide both sides of the equation by 2.
(16)/2 = (2x)/2 :
8 = x
Now we have solved the equation for x. If we go back to the original equation and substitute 8 for x, we will hopefully find that the equality still holds true.
3*(8) + 24 = 8 + 5*(8) :
24 + 24 = 8 + 40 :
48 = 48
This just about wraps up the basics of elementary algebra. In the next part I will go more in depth and develop the basic concepts discussed in this first part. For now, there are a few definitions to understand that are important for every stage of algebra from here on out.

term – Any number or variable, coefficient or not, in an equation or expression. Example (terms are in bold):
3x + 4 = y/2 + 10

variable – A symbol used to represent either a single number or a set of possible numbers. Example: “a,b,c,x,y,z”
expression – A set of numbers, variables, and various arithmetical operations connecting each term. Example “4x²+ 2x + 9”
equation – An equation says that two “things”, either terms or expressions, are equal to each other.
Example: “7 – c = 1 + 2c”