Solving Simultaneous Equations by the Algebraic Method

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:

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.

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 ∴

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

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.

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.

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 = 1200 ……(2)

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 = 1200 ……(2) By substituting (1) into (2), we have x(70  x) = 1200 x2  70x + 1200 = 0 (x  30)(x  40) = 0 x = 30 or x = 40

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.

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?

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 + 3 ……(1) ∵ The product of the two digits is 40. ∴ xy = 40 ……(2)

A digit cannot be negative. y = x + 3 ……(1) xy = 40 ……(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.