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Simultaneous Equations


Simultaneous equations are a set of equations which have more than one unknown values. Questions involving simultaneous equations require students to find the unknowns. First, we have to represent the equations with different numbers or letters for clear explanation. Then we proceed with the below steps.

There are generally two methods to solving simultaneous equations.

•  By substitution

•  By elimination

It may be better to use one method over the other for certain type of simultaneous equations question. Only with practice, will you be able to deduce which method best suits that specific question.


Solve the simultaneous equations 2x + y = 5 and x + 2y = 7

Remember that we first have to represent the equations with proper symbols.

(1) 2x + y = 5

(2) x + 2y= 7

We shall solve it first with the substitution method and show the elimination method at a later stage.

Method of Substitution

In the method of substitution, we express x in terms of y in one equation and substitute it into the other.

From (2) , x = 7 - 2y (3)

Substitute (3) into (1).

2(7-2y) + y = 5

y = 3

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

x = 1

Method of Elimination

Let us learn how to solve simultaneous equations using elimination. Taking the above example, we can choose to eliminate x.

2x + y = 5 (1)

x + 2y = 7 (2)

To eliminate, say, x:

Multiply to obtain the same number of x's on both equations and cancel the x's by subtraction. This leaves y.

2 x (2) : 2x + 4y = 14 (3)

(3) - (1) : 3y = 9

y = 3

Again by substituting y = 3 into (1) or (2),

we arrive at the same conclusion that x = 1.

Both methods are equally suited to the solution of this problem.In any case, the aim is to reduce the two unknowns to one through manipulation of both the equations.

'Try it Yourself' Section

Solve the following simultaneous equation.

3x + 2y = 5 , x + y = 1

Learn more about solving challenging simultaneous quadratic equations

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