This tutorial illustrates and explains how to solve linear equations for x. It also explains the basic intuition involved in thinking about equations. Linear equations are equations with just a plain old variable like “x”, rather than something more complicated like x2 or x/y or square roots. The tutorial only assumes a knowledge of basic arithmetic.
What is an Equation?
An equation is a statement that says that two expressions are equal. This is indicated by using the equals sign (=). The following statement can be thought of as an equation:
5 + 5 = 10,
Read in English this may be read as, “5 plus 5 is the same as 10”.
Alternatively the same statement could have easily been written as:
10 = 5 + 5,
Which in turn read in English translates to, “10 is the same as 5 plus 5”.
Because the equals sign separates the equation into a left hand side and a right hand side, it is possible to think of an equation as a statement that merely states that “the left hand side is the same as the right hand side”. In other words, “the left hand side will evaluate to the same value as the right hand side”.
The most important thing to note at this point is that for a mathematical statement to be an equation the left hand side has to always evaluate to the same value as the right hand side.
How To Solve An Equation for X
Now that you know what an equation is, we will show you how the idea of an equation can be used to solve for unknown values. In general, to solve an equation for a given variable or unknown value, you need to “undo” whatever has been done to the variable. You do this in order to get the variable by itself; in technical terms, you are “isolating” the variable.
Consider the following statement which is also an equation:
5 + x = 10
Read in English this statement translates to, “5 plus some number that we do not know yet is equal to 10”. The first thing you’ll notice is that the above equation is different from the ones we’ve seen till now in that it has a variable x on its left hand side, which is used to indicate a number whose value we do not know yet. It is useful to think of x as a placeholder. With that in mind, often when we are working with equations what we are really interested in finding out is “what is the actual value of the number that x is filling in for in its duty as a placeholder”. We can find this out by solving the equation for x. What that means is that we want to manipulate our equation so that we isolate x to its own side in the equation. The golden rule to solving equations is that whatever you do the left hand side, you have to do on the right hand side in order for the equation to balance out and remain an equation. To solve the above equation for x, we are going to isolate x so that it remains on the left hand side on its own. To do this we subtract 5 from the left hand side, but to follow our golden rule which states that anything you do on the left hand side has to also be done on the right hand side (and vice versa), we will also have to subtract 5 from the right hand side. This will give us the following:
5 + x – 5 = 10 – 5, which simplifies to x = 5. We’ve now solved for x. We’ve shown that x is 5. It’s really that easy.
You will encounter other, more complex equations in your mathematics journey, but the basic ideas for solving an equation are always the same. Your main goal should be to isolate x and have it on its own side of the equation. To isolate x, you will have to perform basic operations like subtraction, addition, multiplication, division and other less basic operations for more advanced equations on the other numbers that appear on the same side of the equation as x. But obviously you need to remember our golden rule that says whatever you do on one side you also have to do on the other side.
Here’s another example:
7 + 3 + x = 12,
Because we want to isolate x, we are going to subtract 7 and 3 from the left hand side of the equation, and we are also going to do the same on the right hand side in keeping with out golden rule. This gives us the following:
7 + 3 + x – 7 – 3 = 12 – 7 – 3,
If we evaluate the above statement step by step, it simplifies to:
10 + x – 10 = 2,
Take note that we did not evaluate the actual numbers along with the x variable that serves as a place older. This is due to another little rule that says that we cannot evaluate unlike terms. We can only evaluate like terms.
1, 2, 3, 5, 8 etc. are like terms. These a just ordinary numbers.
X is unlike the above ordinary numbers, therefore you can’t evaluate x along with the ordinary numbers.
Now to get back to our half solved equation, you can now see that it will evaluate to:
x = 2,
Solving for x gives us 2.
Before we move on to our next example where we solve for x, here is a further note on like terms and unlike terms.
You now know that in an equation like 7 + 3 + x = 12, you have two different types of terms: terms that are ordinary numbers as well as x terms (terms where the variable is x).
But what if on the left hand side you had something life 7 + 3 + x + x, can you say how many different types of terms are in that expression? You are right if you said the above expression has two different types of terms. It can also be evaluated to 10 + 2x, because the ordinary numbers are evaluated on their own and the x’s are evaluated on their own. The x terms are evaluated as follows:
x + x, can be read in English as “an x plus an x”. And so “an x plus an x” simply evaluates to “two x’s”, hence: x + x = 2x
Here is our next example:
3x + 5 = 17,
The first step is to isolate the x term by subtracting 5 from the left hand side and then also subtracting 5 from the right hand side. This will give us the following:
3x + 5 – 5 = 17 – 5, which simplifies to 3x = 12. We now have the x term isolated, but we’re not yet done solving for x.
Read in English 3x = 12 states that, “3 three x’s is the same as 12”, but what we want to find out is the value of one x. To do this we have to get ride of the 3 in front of the x. The best way to remove the 3 from in front of the x is to divide 3x by 3, because we know that 3x/3 = x. The 3’s will cancel out with each other and leave us only with x. We also know that if we divide the left hand side by x, we also have to divide the right hand side by x. So our expression 3x = 12 will now look as follows:
3x/3 = 12/3,
the above expression then goes on to evaluate as follows:
x = 4,
We’ve solved for x and found that x is in fact equal to 4.
Here is another example that looks a little different from the ones we’ve seen thus far:
16 + 4 + 3x = 50 + 2x,
As always the first thing you want to do here is evaluate the like terms on each side of the equation. Doing this will give you the following:
20 + 3x = 50 + 2x,
next you want to bring all your x terms to the same side, evaluate them, then isolate your x.
To bring the 2x on the right hand side to the left hand side, you have to get rid of it from the right hand side by subtracting it from the right, but you obviously also have to perform the same operation of the left hand side. This is what your equation will look like
20 + 3x – 2x = 50 + 2x – 2x, which evaluates to:
20 + x = 50,
Now notice how the x term now only occurs on the left hand side of the equation. The equation now looks very familiar and evaluating it further to solve for x is as easy as:
20 + x – 20 = 50 – 20,
therefore x = 30
Here’s another example which will let you use the two variations we’ve learned so far for isolating x:
6x + 3x = 24 + 3 + 3x,
first we evaluate like terms on each side of the equation:
12x = 27 + 3x,
next we make sure that all x terms are on the same side of our equation by subtracting 3x from both sides of the equation, so that we get rid of the x term on the left hand side:
12x -3x = 27 + 3x -3x, this evaluates to:
9x = 27,
now we have to get x on its own using division to get ride of the 9 in front of the x, but as always, if we divide 9x by 9 on the left hand side, we are going to also have to divide 27 by 9 on the right hand side:
9x/9 = 27/9, this evaluates and simplifies to the following:
x = 3
We have successfully determined that solving for x in the equation 6x + 3x = 24 + 3 + 3x will give us x = 3
At this point you now have the basic intuition involved in solving for x in just about any linear equation that your are confronted with. Of course mathematics has more complex equations than the ones we’ve seen now, but the intuition gained here is the same intuition you will always us when solving equation for x or for any other variables that you may be presented with during your maths journey.
Here are a few practice equation for you to solve on your own in order for you to practice and consolidate your knowledge when it comes to solving for x:
Solve for X in each of the following cases:
1.) x + 7 = 8
2.) 5x + 2x = 27
3.) 7 + 10 + x = 30
4.) x – 4 = 10
5.) x – 2 = 10
6.) 2x – 6 = x + 8
7.) 100x = 1000
8.) 2x + 4 = 36
9.) 9x = 81
10.) 10x = 5x + 100
Please Take Note
As a final note to this tutorial it is important that you stay open minded when looking at equations. Often the variable used in the equations that you will be asked to solve will possibly be y or a or b. In these cases it is important that you realize that the letter or symbol that is used to indicate a variable is not really important and that it has no bearing on how you go about solving an equation.
So for instance the equation x + 7 = 8, is exactly the same as the equation y + 7 = 8 or b + 7 = 8. The only thing that is different is the symbol used to indicate the variable in its role as a placeholder. So you should solve for any other variable you are confronted with exactly the same way as we’ve been solving for x in this tutorial.
Solutions For Exercise
The following are the answers to the questions we asked you to try out on your own:
1.) x = 1
2.) x = 4
3.) x = 13
4.) x = 14
5.) x = 12
6.) x = 14
7.) x = 10
8.) x = 16
9.) x = 9
10.) x = 20
Now that you’ve seen how to solve equations for x, we advise you to solve as many equations as you can. Mathematics is a discipline that is heavily based on practice and consolidation. The more practice you are able to get in, the more consolidated the ideas become and this in turn prepares you for the even more complex ideas that will naturally present themselves as you build upon your maths skills and your knowledge of mathematics.
Non-linear Equation Teaser, where x is raised to the power of 2
After you get a hang of solving these types of equations, the natural progression will be to move on to solving non-linear equations. These are equation in which x is to the power of some number. Non-linear equations will form the bulk of the equations that you will be expected to solve if you decide to follow a career in mathematic or within the sciences. The following is an example of a non-linear equation:
x^2 = 16,
the basic idea here is still to isolate x. To do this you have to evaluate the square-root of x^2, but because what you do on one side of the equation you have to do on the other side, you are required to also evaluate the square-root of 16.
square_root(x^2) = square_root(16)
Evaluating this you will find that square_root(x^2) is x, and that square_root(16) is plus and minus 4. Take note that because x is raised to the power of two this means x has two roots:
x = -4, x = 4
Hope that the above example is enough to pique your interest in learning to solve even more complicated equations.