Exercise 4

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

As we already know, we can solve it numerically by getting x "on its own". When we did this in the earlier example we got x = -1.5. Graphically speaking, we got a straight, vertical line which cut the x-axis at -1.5.

We are now going to see a different way of representing this equation geometrically compared to the previous exercises. 

If we bring all the terms of the equation over to the LHS and simplify it we get:

2x + 3 = 0, which is equivalent to the equation we had at the beginning. It therefore has the same solution.

If we call the LHS of the equation "y" we get the  function y = 2x+3. You may already know that this is called an affine function (a linear function + a constant) and its graph is a straight line, as you can see in the following window.

As you can see, the line cuts the X-axis at a certain point. The "x coordinate" is the point where we find the solution to the equation.

As you can see the solution is x = -1.5.

Move the point along the straight line until it cuts the X-axis. The value given at this point is the solution to the equation.

Therefore, if we bring all the terms of a simple equation over to the LHS we get an expression similar to this one: mx + c = 0.

If we call the LHS of this new equation "y" we get the general equation for an affine function:

y = mx + c

As we have already seen, the graph of an affine function is a straight line which cuts the X-axis at a certain point. The x coordinate of this point is the solution of the initial equation.

In the following window we will see both types of graphical representation that we have focused on so far, and how they give us the same solution.

Exercise 5.

Solve the following equations in your exercise book:

a) 1-3x = 2x - 9
b) x/2 - x/3 = 1

In this example you should try and solve the equation by bringing all the terms over to the LHS and make the RHS (Right Hand Side) equal to 0. This way you will be left with an equation in the form mx+c=0, which is similar to the equation for an affine function y=mx+c. 

For example, you should get -5x+10=0 as the first equation and the corresponding affine function is y=-5x+10.

When you have finished check your results in the following window. Read the instructions carefully next to the window.

There are two boxes at the bottom of the window: in the left-hand box the affine function, which is obtained as described above, is written in red; in the right-hand box you should write the initial equation in blue and press "enter". Use the control keys underneath to introduce the values for m and c in the affine function.  The vertical blue line gives us the solution using the first method explained; the red line gives us the solution using the second method explained. Check that the solution is the same in both cases.

In general, if we bring all the terms over to the LHS of the equation we get an equation which, even without simplifying it, expresses the same affine function as if it were simplified (even though it isn't expressed in the form y = mx+c). For example:

Exercise 6.

Solve: x/2 + x/4 = 3.

If we bring everything over to the LHS we get the equation x/2 + x/4 - 3 = 0 and the function y = x/2 + x/4 -3, which does not need simplifying.

In the following window we can see the graph of the function and the point where it cuts the X-axis: x = 4, which corresponds with the solution of the equation.

In order to alter the equation on this screen just place the cursor on the equation, delete the existing one and write in the new one. (remember that the multiplication symbol is * and the division symbol is / ).