A Tutorial on Solving Equations Part 1

Solving equations is one of the most fundamental concepts that you should learn to be able to solve a lot of mathematical problems such as those in the Civil Service Examinations. For example, for you to be able to solve a word problem, you need to translate words into expressions, set up the equation, and solve it. Therefore, you should learn this post and its continuation by heart.

In this series of posts, we are going to learn how to solve equations and then learn how to solve different types of problems (number, age, coin, Geometry, motion, etc). These types of problems usually appear in the Civil Service Examinations.

So, what is an equation really?

An equation are two expressions (sometimes more) with the equal sign in between. The equation

$2x + 3 = 9$

means that the algebraic expression on the left hand side which is $2x + 3$ has the same value as the numerical expression on the right hand side which is $9$ . Now, you can think of the equal sign as a balance. If you put two different objects and they balance, it means if you take away half of the object on the left, you also have to take half of the object on the left. Or, if you double the amount (or weight) of the object on the left, you also double what’s on the right to keep the balance.

The fancy name of this ‘principle’ in mathematics is Properties of Equality. It basically means that whatever you do on the left hand side, you also do on the right hand side of the equation. Here are a few examples to illustrate the idea.

How to Solve Equations

Example 1: $x + 4 = 9$

There is really nothing to solve in this example. What will you add to $4$ to get $9$ . Of course $5$ . However, we use the Properties of Equality future reference. The idea is to isolate $x$ on one side and all the other numbers on the other side. Since, $x$ is on the left hand side, we want to get rid of $4$ . So, since $4$ was added to $x$ , we have to subtract $4$ from both sides to get rid of it. So,

$x + 4 - 4 = 9 - 4$ .

This gives us $x = 5$ .

Example 2: $3x = 18$

This example can be again solved mentally. What will you multiply with $3$ to get $18$ , of course, it’s $6$ . But, solving it as above, to get rid of 3 in $3x$ , since it is multiplication, we divide it by $3$ .

$\displaystyle \frac{3x}{3} = \frac{18}{3}$ .

Of course, if you divide the left hand side by $3$ , you also divide the right hand side of the equation by $3$ .

This gives us $x = 6$ .

Example 3: $\frac{x}{5} = 12$

In this example, $\frac{x}{5}$ is a fraction which mean that we have to get rid of 5. To do this, we multiply both sides by $5$ . That is,

$5(\frac{x}{5}) = 5(12)$

Therefore, $x = 60$ .

Like Examples 1 and 2, this can be solved mentally.

Example 4: $2x + 3 = 9$

In this example, we have 2 times $x$ and then added to $3$ . Well, intuitively, we can eliminate $3$ first by subtracting it from both sides. That is

$2x + 3 - 3 = 9 - 3$

which results to

$2x = 6$ .

Now, it’s multiplication, so we eliminate $2$ by on the left hand side by dividing both sides by $2$ . That is

$\displaystyle \frac{2x}{2} = \frac{6}{2}$ .

This results to $x = 3$

Example 5: $4x - 4 = 9$ .

We first need to eliminate $4$ on the left hand side of the equation. Since it is subtraction, to eliminate it, we have to perform addition (because $-4 + 4 = 0$ ) on both sides of the equation. Doing this, we have