SIMULTANEOUS LINEAR AND QUADRATIC EQUATIONS

CONTENT

1. Simultaneous linear equations (Revision).
2. Solution to linear and quadratic equations.
3. Graphical solution of linear and quadratic equations.
4. Word problems leading to simultaneous equations.
5. Gradient of curve.

SIMULTANEOUS LINEAR EQUATIONS (REVISION)

Simultaneous linear equations (revision)

Recall: Simultaneous means happening or done at the same time i.e. following each other. It can be solved either graphically or algebraically. Algebraically involves using either substitution or elimination methods

Example 1: Solve the following pair of simultaneous equations

\(2x -y = 8 \\ 3x + y = 17\)

Solution:

ELIMINATION METHOD

(a) \(2x -y = 8,\text{ } 3x + y = 17\)

On adding both equations, we have

\(5x = 25 \\ x = 5\)

Substituting for \(x\) in equation (1)

\(2x -y = 8 \\ 2(5) – y = 8 \\ y = 2 \\ ⇒ x = 5, y = 2\)

Using substitution method;

(a) \(2x -y = 8\)………(1)

\(3x + y = 17\)………(2)

In (2), \(y = 17 -3x\)………(3)

Substituting in equation (1), we have

\(2x -(17 -3x) = 8 \\ 2x -17 + 3x = 8 \\ 5x = 25 \\ x = 5\)

In (3) \(y = 17 -3(5) \\ = 17 -15 \\ 7 = 2\)

EXAMPLE 2: Solve the pairs of equations:

\(\frac{27^x}{81^{x +2y}} = 9 \\ x + 4y = 0\)

Solution:

\(\frac{3^{3x}}{3^{4(x +2y)}} = 3^2 \\ x + 4y = 0\)

Recall, \(\frac{2}{a^2} = \frac{2}{1} × \frac{1}{a^2} = 2a^{-2} \\ ⇒ 3^{3x} × 3^{-4(x + 2y)} = 3^2 \\ 3^{3x -4(x + 2y)} = 3^2\)

Comparing the powers of 3, we have,

\(3x -4(x + 2y) = 2 \\ 3x -4x -8y = 2 \\ -x -8y = 2 ……..(i) \\ x + 4y = 0 ………(ii)\)

Solving equations (i) and (ii) we obtain values for x and y

i.e.