Descripción.

	Simple equations: An explanation.
	3rd year of secondary education.

Explanation and examples.

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 :

1 - 3x = 2x - 9

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

x - 3 = 2 + x

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

In this unit we are going to look at different ways of solving these simple equations with an unknown factor.

Numerical and graphical solutions.

Exercise 1

Imagine that we want to solve the equation: 3(x-1) = 4 - 2(x+1).

As you already know, 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 = 2 gives us 3 = -2, which is not true, so 2 is not a solution.

x = 1 gives us 0 = 0, which is true. Therefore we have found a solution for the equation. Later on we will see how sometimes there can be more than one solution.

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. This involves carrying out certain operations and bringing terms over from one side to the other until x = a number. Therefore, going back to the example:

3x - 3 = 4 - 2x - 2 (careful when there's a sign before the brackets)

3x + 2x =3 + 4 - 2 ; 5x = 5; x = 5/5 ; x = 1 which is the solution we found earlier.

In this example we have found solved the equation numerically. Now we are going to see how to solve the equation graphically.

The equation that we want to solve appears in red in the lower part of this window. The red, vertical line shows the solution of the equation.

The solution or root of the equation is the point where this straight line cuts the X-axis.

Find the solution by moving the red point with the mouse until it reaches the vertical line.

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. The following exercise shows us how this is done:

3x + 1 = x - 2.

a) 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".

b) 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 which is the solution. 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".

The following can occur in more complex equations:

- Certain operations are indicated by brackets, which we have to carry out first (as we did in exercise 1).

- There are denominators in the equation. In this case find a common denominator for both sides and simplify the equation before continuing as above (see exercise 3).

For example: In order to solve the equation:

and get rid of the denominators, which leaves us with what we had in the last example: 2(x - 2) - 3(x + 3) = 5(1 - 2x)

Exercise 2.

In your exercise book solve the following equation numerically:2(x-5) = -2(x-3). Check your solution in the following window by following the instructions given in the box on the right.

In the following window write the equation from this exercise on the line where the equation from the previous exercise is written. (Note that you should use the* symbol to express a product and be careful with the brackets as they may be necessary even though they're not written out).
Click on the enter button and the solution to the equation should appear as a red, vertical line. To find the solution move the red point in the centre with the mouse until it reaches this vertical line. The value you get should be the same as the one you have in your exercise book.

Exercise 3.

Solve the following equation in your exercise book: x/2 + x/3 = 5. Then use the window above to check your results.

	Leoncio Santos Cuervo

	Spanish Ministry of Education. Year 2001

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