The first thing to grasp is that when we have an equation, both
sides have exactly the same value.
Let's start with:
8 = 8
That is an equation. Simple enough? Now we change the equation a little by introducing simple arithmetic operations that you already know:
5 + 3 = 8Thus: 5 + 3 = 2 × 4
8 = 2 × 4
Easy to follow so far? OK, the next step is something you may done in arithmetic quizzes in grade school:
5 + ☐ = 2 × 4
If you are asked to fill in the box, you can do the simple arithmetic and know that the answer should be 3. Now we are ready for basic algebra. Let's substitute the box with the letter 'k' and we have:
5 + k = 2 × 4In the equation above, the letter 'k' is known as a variable. Of course we know that it is 3, so why is it called a variable? Well, that's the way algebra is - there are just some terms where the meaning is not as straightforward. You may think of it this way - if you were just given the equation 5 + k = 2 × 4 without any of the earlier discussions, then k would be unknown until you solve the arithmetic. That's the idea for variables in algebra. Anyway, variables are defined as numbers that can change value or represent a missing value (an unknown value). Variables are usually represented by letters of the alphabet, and the letters x, y, and z are most commonly used.
Now we have a real basic algebra equation, and the goal is to solve for the variable k - that means to find the value of 'k' in the equation. Of course we know from earlier our earlier exercises that k = 3, but hey, where's the fun if algebra is just like that?
So, an algebra equation would be given as: 5 + k = 2 × 4 without any of the earlier exercises and you would be asked to solve for the unknown k.
Before we go about solving for the variable k, there's just one simple principle of equations that we need to grasp. Since we know that both sides of the equation are the same, whatever we do on one side (arithmetically), if we do the same to the other side, and the result is still an equation - that means both sides would still be equal. For example, we can do any of these:
5 + k - 2 = 2 × 4 - 2
5 + k + 4 = 2 × 4 + 4
(5 + k) × 3 = (2 × 4) × 3
Now we are ready to tackle our first algebra equation. What we want to do is to isolate the variable k on one side of the equation. Let's start with the equation:
5 + k = 2 × 4We can see that on the left side, there's an extra 5 added to k. So we must get rid of the 5 to isolate k. We can do this be subtracting 5 from the left side. Remember that we must do the same thing to the right side to maintain equality:
5 + k - 5 = 2 × 4 - 5Now we are almost done solving our first algebra equation! Looking at the left side 5 + k - 5, the two 5s (5 and -5) would cancel out, leaving us with:
k = 2 × 4 - 5So we only need to do the arithmetic on the right side:
k = 2 × 4 - 5Voila! We have solved our first algebra equation! Remember, the goal is to get the variable alone by doing the same thing to each side of the equation. With this you have a good understanding of basic algebra, and now you should be able to solve other equations like 6 + k = 11 or 11 - m = 7. Otherwise, you may want to re-read this lesson.
k = 8 - 5
k = 3
Just one more simple thing to finish up. In algebra you would often see something like 6k or 14m used in equations. They just mean 6 × k and 14 × m - just think of it as a mathematician's shorthand. You can figure out why they prefer to omit the × sign especially when the letter x is most commonly used as the variable in algebra equations.
If you are comfortable with the basic algebra in this lesson, you are now ready to go to Isolate the Variable (Transposition).
You may also want to practice with some basic algebra worksheets. Basic Algebra Worksheets.
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