1. More about Equations (1)11
More about Equations (1)
Case Study
11.1 Solving Simultaneous Equations by the Graphical Method
Chapter Summary

2. Case Study
How can we find the position where the ball lands on the inclined plane?
You have to know the path of the ball first.
When the ball left the starting platform, its flying path can be represented by a parabola.
The inclined plane can be represented by a straight line.
The intersection of the parabola and the straight line is where the ball lands on the inclined plane.

3. 11.1 Solving Simultaneous Equations by the Graphical Method
A. Solving Simultaneous Equations in Two
Unknowns, One Linear and One Quadratic
In junior forms, we learnt that a pair of simultaneous linear equations in two unknowns can be solved graphically.
We first draw the graph of each of the equations on the same coordinate plane.
The point of intersection of the graphs is the solution of the two equations.
For example, the solution of the simultaneous
linear equations is (4.0, 5.0).
Similarly, when we need to solve a pair of simultaneous equations in two unknowns in which one is linear and one is quadratic, we can use the same method to obtain the solutions.
The solution may also be
written as

4. Example 11.1T 11.1 Solving Simultaneous Equations
by the Graphical Method
A. Solving Simultaneous Equations in Two
Unknowns, One Linear and One Quadratic
Example 11.1T
Solve the following simultaneous equations graphically.
Solution:
For y  x2,
y  2x  3 x 3 2 1 1 2 3 y 9 4
For y  2x  3, x 3 2 1 1 y 9 7 5 3
From the graphs, the points of intersection are (3.0, 9.0) and (1.0, 1.0).
∴ The required solutions are (3.0, 9.0) and (1.0, 1.0).

5. Example 11.2T 11.1 Solving Simultaneous Equations
by the Graphical Method
A. Solving Simultaneous Equations in Two
Unknowns, One Linear and One Quadratic
Example 11.2T
Solve the following simultaneous equations graphically.
Solution:
y  4x  17
For y  x2  2x  8, x 2 1 1 2 3 4 y 5 8 9
For y  4x  17, x 2 3 4 y 9 5 1
From the graphs, the only point of intersection is (3.0, 5.0).
∴ The required solution is (3.0, 5.0).

6. Example 11.3T 11.1 Solving Simultaneous Equations
by the Graphical Method
A. Solving Simultaneous Equations in Two
Unknowns, One Linear and One Quadratic
Example 11.3T
Solve the following simultaneous equations graphically.
Solution:
For y  x2  3x  4, x 5 4 3 2 1 1 y 6 6
For y  4x  20, x 5 4 3 y 8
y  4x  20
From the graphs, there are no points of intersection.
∴ There are no solutions.

7. 11.1 Solving Simultaneous Equations by the Graphical Method
A. Solving Simultaneous Equations in Two
Unknowns, One Linear and One Quadratic
From the above examples, we can see that there are three different possible outcomes when solving simultaneous equations in two unknowns (one linear and one quadratic) graphically:
• Two distinct solutions
• Only one solution
• No real solutions
The number of solutions depends on the number of points of intersection of the graphs of the two equations.
y  5x  44
In Book 4 Chapter 2, we learnt that the solutions found by the graphical method are approximations only.
As shown in the figure, if the scale of the gridlines are not small enough, it is difficult to read the points of intersection.

8. 11.1 Solving Simultaneous Equations by the Graphical Method
B. Applications of the Graphical Method
Graphical method can also be used to solve real-life problems, as illustrated in Example 11.4T.

9. Example 11.4T 11.1 Solving Simultaneous Equations
by the Graphical Method
B. Applications of the Graphical Method
Example 11.4T
The figure shows a rectangular sheet with dimensions 8 cm  5 cm. Four squares at the corners, each of side x cm, are cut off as shown in the figure.
(a) Let y cm2 be the area of the shaded region.
Express y in terms of x.
(b) The figure shows the graph of y  4x2  40. Using the graph, find the value of x if the area of the shaded region is 20 cm2.
(Give the answer correct to the nearest 0.2.)
Solution:
(a) Area of each square  x2 cm2
Area of the shaded region  (8  5  4x2) cm2
∴ y  40  4x2
y  20
(b) ∵ Area  20 cm2
∴ Draw the line y  20 on the graph.
∴ x  2.2 (cor. to the nearest 0.2)
or 2.2 (rejected)

10. Chapter Summary 11.1 Solving Simultaneous Equations by the
Graphical Method
1. The points of intersection of the line
y  px  q and the parabola y  ax2  bx  c are the solutions of the simultaneous equations:
From the above figure, the solutions of the simultaneous equations are (x1, y1) and (x2, y2).
When solving simultaneous equations graphically, there are three possible cases, that is, two distinct solutions, only one solution or no real solutions.
2. The graphical method can also be used to solve real-life problems.