Solving linear and non-linear simultaneous equations

Concept

The method of elimination may not always be possible for simultaneous quadratic equations. This is because the terms of the unknown variable may not always cancel out accordingly. Thus, in such cases, we have to rely on the method of substitution to solve these simultaneous equations.

Example

Solve the simultaneous equations

Like always, we have to represent the equations with symbols for proper presentation.

x + y = 5 (1)

$pic showing simultaneous equation$ (2)

Normally, we would take the linear equation and express an unknown variable in terms of the other.

From (1) ,

Substitute (3) into (2).

Afterwards, we would substitute it into the quadratic equation and solve.

$simultaneous equation that is being expanded$

(y-2)(y-3)= 0

y = 2 or y = 3

Substitute y = 2 and y = 3 into (1).

When y = 2 , x = 3

When y = 3 , x = 2

Here, substitution is necessary to solve the simultaneous equations. By using the first equation, we are able to substitute one unknown in a relatively simple form of the second unknown, as opposed to using the second equation for substitution purposes. Note that because quadratic powers are involved we obtain two sets of solutions for this pair of simultaneous, non-linear equations.

'Try it Yourself' Section

Try solving simultaneous quadratic equations below.

a) $simultaneous equation to solve$ ,x = y + 4

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