Graphical solution of equations
Equations of first degree in one unknown
Consider the equation 2x - 3 = 0. Substitute for x arbitrary
values and denote the results each time by y, you then
have the function y=2x-3, the image of which is the
straight line shown. If you want to determine the root of this equation graphically,
you must find the point, where y = 0. Hence it must lie on
the abscissa and be the point
of intersection of the line with the abscissa. You will see from the drawing
that y = 0 for x = 1.5, that is, the root of the
equation 2x-3=0 is x = 1.5.
Example: Solve graphically the equation
4(x+1) - 13 = 5(x - 1) - 3x.
Move all terms to one side:
0 = 5(x - 1) - 3x - 4(x + 1) + 13.
The function is
y = 5(x - 1) - 3x - 4(x + 1) + 13.
You see that you can give it the form y=mx+n, but you need not do so!
For x = 1, y = 2, for x = -1, y = 6.
The line passes through the points P1(1,2) and P2(-1,6); the root is the abscissa of the point S: x = 2.
Equations of the first degree with two unknowns
Task: For the
equations I: y=x/2+1.5,
and II: y = 4x-2, find graphically
(x,y) which satisfies both equations!
There are any number of pairs of values which satisfy the first equation, for example: (2,1.5) or (-1,+1), they are coordinates of points on Line I. Similarly, you find on Line II the points (+1.5,+4) and (0,-2). They lead you to points on Line II.
Since the point of intersection S lies on both lines, its coordinates (+1,+2) must satisfy both equations. Let us test and set x = 1, y = 2 in the equations of the two lines. You see that both are satisfied.
Since two straight lines can only
have one point of intersection, there can only be one pair of
values, which satisfies both equations. This pair of values is
called the root of the given equations.
Task: Find graphically the root of the equations:
| I: | (x-3)/2+(y-1)/4 = -1 | II: | (y+2x)/2-(x-1)/2= 7/4 |
| y | x | y | x |
| -3 | 3 | 1.5 | 1 |
| +3 | 0 | 2.5 |
You realize immediately that you are dealing with linear equations, so that their images are the straight lines shown. The root is x = 0.5, y = 2.
For drawing a line, you should not select two points which lie too close together, because your drawing will become inaccurate! You can check the accuracy of your drawing by computing x and y for a third point and measuring the corresponding ordinate. If for the chosen scale the point of intersection lies outside your graph paper or the graphical result for the root is not accurate enough, you change the scale of a partial drawing for better results.
Task: Find graphically the root of
the equations I: y =3x+5 and II: y=8x-20.
Use the pairs of points (y,x): I: (-1,-2)
and (5,0), II: (-4,2) and (4,3). 
Once you know approximately the
location of the root, you can draw the area around it in a larger
scale; as a rule, you must then use other points and recalculate
values.
Task: Two motor cars head for each other at the same time from A at 60 km/hr and from B, 75 km away, at 75 km/hr. How far from A will they meet and when?
Enter the distance from A
on the ordinate axis in the scale 10 km 
5 mm, the time on the
abscissa at 10 minutes 1/2 cm.
Car I departs from P1(x1
= 0, y1= 0) and is in 60 minutes 60 km from A
at P2(x2 = 60, y2=
60). P1 and P2 determine the Line I.
Car II, when it departs, is
75 km from A at
P(x0 = 0, y0
=75) and after 60 minutes, at 65 km/hr, 10 km from A at P4(x4=60,y4=10).
P0 and P4 determine Line II.
You can now extract from the drawing the required answer. The point S lies at (36,36). The cars meet after 36 minutes at a distance of 36 km from A.
The drawing lets you determine a number of other relations; for example, after 12 minutes, Car I is 12 km, Car II 62 km from A or, after 24 minutes, the cars are 25 km apart or Car I arrives at B after 75 minutes, etc.
Exercises:
| 1. | An aeroplane flies 360 km against the wind in 1hr and 40 mins, it returns with the same wind in 1hr 30 mins.How fast does it fly and what is the velocity of the wind? | |
| 2. | Father, mother, daughter and son together are 96 years old. Father and son together are as old as mother and daughter. Father and daughter together were three years ago as old as mother and son are together today. Two years ago, father and mother together were tree times as old as daugher and son today. How old is every one today? | |
| 3. | If you extend the short sides of a right-angled triangle by 2 cm, its area increases by 14 cm². If you shorten them by 2 cm, the area drops by 10² cm. How long are these sides? |