Equality in Algebra

If you’ve ever struggled with algebra, wondered which operations you were allowed to perform, or been upset when you were told that the operation you just performed was incorrect, this post is for you. Algebra is all about equality.
What does that mean?

Consider the statement:

8 = 8

That’s obviously not groundbreaking news, but it does show the underpinnings of what makes algebra “work.” When you use that equals sign, =, you’re saying that what’s on the left of that sign is the exact same value as what’s on the right hand side of that sign. 8 = 8 means “8 is the same exact value as 8.” And then as long as you do the same thing to both 8s, you’ll preserve that equality. So you could subtract 2 from both sides:

8 – 2 = 8 – 2

And you’ll arrive at another definitely-true statement:

6 = 6

And then you could divide both sides by 3:

6/3 = 6/3

And again you’ve created another true statement:

2 = 2

Because you start with a true statement, as proven by that equals sign, as long as you do the exact same thing to what’s on either side of that equals sign, the statement will remain true. So when you replace that with a different equation:

3x + 2 = 8

That’s when the equals sign really helps you. It’s saying that “3x + 2” is the exact same value as 8. So whatever you do to that 8, as long as you do the same exact thing to the other side, the equation will remain true. Following the same steps, you can:

Subtract two from both sides:

3x = 6

Divide both sides by 3:

x = 2

And you’ve now solved for x. That’s what you’re doing with algebra. You’re taking advantage of that equality: the equals sign guarantees a true statement and allows you to do the exact same thing on either side of that sign to create additional true statements. And your goal then is to use that equals sign to strategically create a true statement that helps you to answer the question that you’re given.

Equality applies to all terms; it cannot single out just one individual term.

Now, where do people go wrong? The most common mistake that people make is that they don’t do the same thing to both SIDES of the inequality. Instead they do the same thing to a term on each side, but they miss a term. For example:

(3x + 5)/7 = x – 9

In order to preserve this equation and eliminate the denominator, you must multiply both SIDES by 7. You cannot multiply just the x on the right by 7 (a common mistake); instead you have to multiply everything on the right by 7 (and of course everything on the left by 7 too):

7(3x + 5)/7 = 7(x – 9)

3x + 5 = 7x – 63

Then subtract 3x from both sides to preserve the equation:

5 = 4x – 63

Then add 63 to both sides to preserve the equation:

68 = 4x

Then divide both sides by 4:

17 = x

The point being: preserving equality is what makes algebra work. When you’re multiplying or dividing in order to preserve that equality, you have to be completely equitable to both sides of the equation: you can’t single out any one term or group. If you’re multiplying both sides by 7, you have to distribute that 7 to both the x and the -9. So when you take the CAT, do so with equality in mind. The equals sign is what allows you to solve for variables, but remember that you have to do the exact same thing to both sides.