Exercise 7

Solve the following equations in your exercise book:

a) x-3 = 2+x.

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

In the first example you should find that 0 = 5 and in the second that 0 = 2. What does this mean? Obviously both equations cannot be true, regardless of the value of x. 

We can say that in such cases the equation doesn't have a solution.

If we make the RHS of both equations equal to 0 and we simplify them as much as possible we get -5 = 0 and -2 = 0. As you can see, x "disappears" which is the same as saying that in the equation mx + c= 0, "m = 0".

In this window we can see the position of the straight line when m = 0, c = -5, as in equation a).

Change the value of c  to any number except 0 and maintain the value of m at 0.

The straight lines are always parallel to the X-axis in all cases! Therefore, the line doesn't cut the X-axis. This means that:

there is no solution to the equation

Exercise 8

Solve the following equations in your exercise book:

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

b) x/2 - x/3 = x/6

Prove it by substituting whichever value you want for x into either of the equations.

Go back to the window above and put in the following values: a = 0 y b = 0. The straight line which is produced coincides exactly with the X-axis so we can therefore say that it "cuts" the X-axis at infinite points, i.e. for all values of x.

In this case we can say that the equation has:

infinite solutions or that it is an identity.

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

Exercise 9

Form a new equation and solve the following problem both numerically and graphically, if you wish, in this window:

Exercise 10

There are 20 questions in an exam. Each question answered correctly is awarded 3 marks and 2 marks are lost for each question answered incorrectly. If a student gets a total of 30 marks in the exam how many questions did he answer correctly and how many incorrectly?

Solve the following equations and problems in your exercise book. Then, check your answers in the following window.

If you are not able to see the straight line at any point change the "scale" until you can see it clearly. "Use the arrows to change the value of x until you find the point where it cuts the X-axis. (Solution)" "You can use either of the two geometric methods presented in this unit: either write the equation directly into the window (in blue) or find the corresponding affine function which has the form y=mx+c, and change the values of m and c in the corresponding box."

Exercise 11

Solve the following equations:

• 2(x - 7) - 3(x + 2) + 4(x + 1) - 2 = 0
• 3x - 5 = x/2
• (x - 9) / 2 = (1 + 3x) / 13
• 
• x/2 - 4 = (x-1)/2 - 7/2

Exercise 12

Solve the following problems:

• A rectangular garden has a perimeter of 58m. If the longest side is 11m longer than the shortest side how long are the sides of the garden? 

• 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.

• One tap takes 3 hours to fill a container with water and another tap takes 6 hours. If the container is empty and both taps are turned on together how long will it take to fill the container?