An equation is made up of two algebraic expressions which include numbers and letters (unknown factors) and are equal to each other (separated by an equals sign).

For example: 3x - 2y = x² + 1

An equation with one unknown factor only has one letter (the unknown is usually represented by the letter x).

For example: x² + 1 = x + 4

An equation is called a simple equation when this unknown is not raised to any power (i.e. it is raised to the power of 1).

Examples:

3x + 1 = x - 2

1 - 3x = 2x - 9.

x - 3 = 2 + x.

x/2 = 1 - x + 3x/2

In this unit we are going to look at different ways of solving these example equations.

Exercise 1 - Imagine that we want to solve the equation: 3x + 1 = x - 2.

In order to solve an equation we need to find a value for x, which when substituted back into the equation satisfies it (i.e. both sides are equal to each other).

We can try putting a value for x into the example:

x =1 gives us 4 = -1, which is not true.

x = -1 gives us -2 = -3, which is not true either. Therefore we have to try and solve the equation in order to find the value of x:

As you already know, in order to solve an equation numerically we first need to get x on its own on one side of the equation so that we get: x = a number. Therefore:

3x - x = -1 - 2 ; 2x = - 3 ; x = -3/2 or x = -1.5

Effectively: 3(-1.5) + 1 = -1.5 -2 ; -4.5 + 1 = -3.5. Which is true!

In this example we can say that the equation has a solution. But:

What does this solution look like graphically?

Look carefully at the following window. The straight red line shows the graph of the equation.

The solution or root of the equation is where the straight line cuts the X-axis (note that x = -1.5)

Change the values for x in the window by "dragging" the large red point with the mouse. Note that the equation is written in red in the lower part of the window.

In order to solve a simple equation we have to follow a few basic rules to get "x" on its own on the LHS (Left Hand Side) of the equation. Let's use the previous example to see how this is done:

3x + 1 = x - 2.

- Add or subtract the same number to each side of the equation. If you subtract 1 and x from each side you get:

3x +1 -1 - x = x - x - 2 -1, which when simplified gives us: 2x = -3. We get the same answer if we "change the sign (+ to - or - to +) when we take terms over to the other side of the equation".

- Multiply or divide both sides of the equation by the same number. In this case by 2:

2x/2 = -3/2, which when simplified gives us x = -3/2 as we got in the previous example. We get the same answer by doing the following: "when you take a factor over to the other side of the equation divide what the other side is multiplied by or multiply what the other side is divided by".

Exercise 2.

In your exercise book solve the following equation numerically:

1 - 3x = 2x - 9.

Check where the straight line cuts the X-axis. The value for x should be the same as that obtained numerically.

In the following window write the equation from this exercise on the line where the equation from the previous exercise is written. Look carefully at how the equation from exercise 1 expresses 3x as 3*x.

Exercise 3. - Solve the following equation in your exercise book:

x - 3 = 2 + x.

You should find pretty quickly that 0 = 5 but what does this mean? This equation is not true for any value of x. Therefore we can say that the equation has no solution.

You will see that there is no straight line in the following diagram. There is no graph for the equation and therefore the X-axis is not cut by a straight line which means there is no solution. 

Exercise 4.- Solve the following equation numerically and show that the equation does not have a solution:

3x - 2 + x = 5x + 1 - x

Change the equation in the window for the one in this last example. Notice how there is no straight line to represent the equation, which therefore does not have a solution.

Exercise 5.-Solve the following equation in your exercise book:

2x-1 = 3x + 3 - x - 4

You should have found that 0 = 0 but what does this mean now? Both sides of the equation are equal to each other but we have got rid of x. What is the solution of the equation?

If both sides of the equation are equal then they will be for any value of x! Prove it by substituting different values for x into the equation e.g. x = 0, 1, -3 or whichever value you want.

In this case we can say that the equation has infinite solutions (any value of x is a solution).

We cannot draw a graph of the equation as we did for the previous examples as the program will not draw a graph for the equation 0 = 0.

These type of equations are known as IDENTITIES.

Exercise 6.- Prove that the following equation is an identity in your exercise book.

3x -2 + x = 1 + 4x - 3

One of the most important uses for these equations is to solve everyday problems. For instance:

Exercise 7.- There are three brothers in a family. The eldest brother is 4 years older than the middle brother, who is 3 years older than the youngest brother. When their ages are added together they are as old as their father, who is 40. How old is each brother?

In order to solve these kind of problems we have to find an unknown factor and call it "x". In this example we shall call x:

the age of the youngest brother.

Then we write an equation using all the information included in the problem. Therefore:

The age of the middle brother = x + 3;

The age of the eldest brother = x + 3 + 4 = x + 7.

The equation: The sum of the brothers' ages = 40; x + x+3 + x+7 = 40,

If we solve the equation we get x = 10, therefore the solution to the problem is:

The brothers are 10, 13 and 17 years old.

You can also see the solution in the following window.

Form an equation and solve the following problem both numerically and graphically in this window:

Exercise 8.- In a box there are twice as many mint sweets as strawberry ones and three times as many orange sweets as mint and strawberry sweets together. If there are 144 sweets altogether how many sweets of each flavour are there? (Sol: 12, 24, 108).

Solve the following problems numerically in your exercise book and graphically in the following window:

Exercise 9.- Solve the following equations:

a) -5x = 12 - x
b) 2(x-7)-3(x+2)+4(x+1)-2 = 0 (Take care with the signs outside the brackets!)
c) 3x - 5 = x/2 (Remember that to get rid of the 2 you can multiply the whole equation by 2)
d) 3x + 4 - x = 7 + 2x
e) 2x - 1 = 3(x + 2) - x

b) Find a number which when you divide by two, add a quarter of the number to this and then add 1 you get the original number. (Sol:4).