1. Solving Simultaneous Equations by the Algebraic Method

2. How to solve ?
y = 3  x
y = x2 + 2x  7
y = 3  x
y = x2 + 2x  7
We can solve simultaneous equations, one linear and one quadratic, by the algebraic method as follow:

3. Eliminate the unknown y to obtain a quadratic equation in x only.
Solve the following simultaneous equations.
y = 3  x ……(1)
y = x2 + 2x  7 ……(2)
Step 1
Substitute the linear equation into the quadratic equation.
By substituting (1) into (2), we have
3  x = x2 + 2x  7
x2 + 3x  10 = 0
Eliminate the unknown y to obtain a quadratic equation in x only.

4. The solutions can also be expressed as:
Solve the following simultaneous equations.
y = 3  x ……(1)
y = x2 + 2x  7 ……(2)
Step 2
Solve the quadratic equation in one unknown obtained.
The solutions can also be expressed as: or
x2 + 3x  10 = 0
(x  2)(x + 5) = 0
x = 2 or x = 5
x = 2
y = 1
x = 5
y = 8
By substituting x = 2 into (1), we have
y = 3  (2) = 1
By substituting x = 5 into (1), we have
y = 3  (5) = 8 ∴

5. Follow-up question
Solve the following simultaneous equations.
x = y2 – 3y + 5 ……(1)
x – 2y – 1 = 0 ……(2)
From (2), we have
x = 2y ……(3)
By substituting (3) into (1), we have
Make one variable the subject of the linear equation before substitution.

7. I am going to show you in the following example.
What are the key steps in solving practical problems leading to simultaneous equations?
I am going to show you in the following example.

8. Represent the unknown quantities by using letters.
The perimeter of a rectangular flag is 140 cm and its area is 1200 cm2. Find the dimensions of the flag.
y cm
x cm
Step 1
Represent the unknown quantities by using letters.
Let x cm and y cm be the length and the width of the flag respectively.

9. Set up equations based on the given conditions.
The perimeter of a rectangular flag is 140 cm and its area is 1200 cm2. Find the dimensions of the flag.
y cm
x cm
Step 2
Set up equations based on the given conditions.
Perimeter of the flag: 2(x + y) = 140
x + y = 70
y = 70  x ……(1)
Area of the flag: xy = ……(2)

10. Solve the simultaneous equations obtained.
The perimeter of a rectangular flag is 140 cm and its area is 1200 cm2. Find the dimensions of the flag.
y cm
x cm
Step 3
Solve the simultaneous equations obtained.
y = 70 – x ……(1)
xy = ……(2)
By substituting (1) into (2), we have
x(70  x) = 1200
x2  70x = 0
(x  30)(x  40) = 0
x = 30 or x = 40

11. Solve the simultaneous equations obtained.
The perimeter of a rectangular flag is 140 cm and its area is 1200 cm2. Find the dimensions of the flag.
y cm
x cm
Step 3
Solve the simultaneous equations obtained.
By substituting x = 30 into (1), we have
y = 70  (30) = 40
By substituting x = 40 into (1), we have
y = 70  (40) = 30
∴ The dimensions of the flag are 30 cm  40 cm.

12. Follow-up question
A two-digit positive integer is increased by 27 when its digits are reversed. The product of the two digits is 40. What is the original integer?

13. Let x be the tens digit and y be the units digit of the original integer.∴
The original integer is 10x + y,
and the integer becomes 10y + x when the digits are
reversed.
∵ The integer is increased by 27 when its digits are reversed.
∴ (10y + x)  (10x + y) = 27
9x + 9y = 27
x + y = 3
y = x ……(1)
∵ The product of the two digits is 40.
∴ xy = ……(2)

14. A digit cannot be negative.
y = x ……(1)
xy = ……(2)
By substituting (1) into (2), we have
x(x + 3) = 40
A digit cannot be negative.
x2 + 3x  40 = 0
(x  5)(x + 8) = 0
x = 5 or x = 8 (rejected)
By substituting x = 5 into (1), we have
y = (5) + 3 = 8
∴ The original integer is 58.