WHAT ARE SIMULTANEOUS EQUATIONS? (EXERCISES SOLVED) - MATHS - 2025

The simultaneous equations are those equations that must be met at the same time. Therefore, to have simultaneous equations you must have more than one equation.

When you have two or more different equations, which must have the same solution (or the same solutions), it is said that you have a system of equations or it is also said that you have simultaneous equations.

When we have simultaneous equations, it can happen that they do not have common solutions or have a finite quantity or have an infinite quantity.

Simultaneous equations

Given two different equations Eq1 and Eq2, it follows that the system of these two equations is called simultaneous equations.

The simultaneous equations satisfy that if S is a solution of Eq1 then S is also a solution of Eq2 and vice versa

characteristics

When it comes to a system of simultaneous equations, you can have 2 equations, 3 equations or N equations.

The most common methods used to solve simultaneous equations are: substitution, equalization, and reduction. There is also another method called Cramer's rule, which is very useful for systems of more than two simultaneous equations.

An example of simultaneous equations is the system

Eq1: x + y = 2

Eq2: 2x-y = 1

It can be seen that x = 0, y = 2 is a solution of Eq1 but it is not a solution of Eq2.

The only common solution that both equations have is x = 1, y = 1. That is, x = 1, y = 1 is the solution of the system of simultaneous equations.

Solved Exercises

Next, we proceed to solve the system of simultaneous equations shown above, through the 3 mentioned methods.

First exercise

Solve the system of equations Eq1: x + y = 2, Eq2 = 2x-y = 1 using the substitution method.

Solution

The substitution method consists of solving for one of the unknowns in one of the equations and then substituting it in the other equation. In this particular case, we can solve for "y" from Eq1 and we obtain that y = 2-x.

Substituting this value of «y» in Eq2, we obtain that 2x- (2-x) = 1. Therefore, we obtain that 3x-2 = 1, that is, x = 1.

Then, since the value of x is known, it is substituted in "y" and we obtain that y = 2-1 = 1.

Therefore, the only solution to the system of simultaneous equations Eq1 and Eq2 is x = 1, y = 1.

Second Exercise

Solve the system of equations Eq1: x + y = 2, Eq2 = 2x-y = 1 using the matching method.

Solution

The matching method consists of solving for the same unknown in both equations and then matching the resulting equations.

Solving for "x" from both equations, we obtain that x = 2-y, and that x = (1 + y) / 2. Now, these two equations are equated and we obtain that 2-y = (1 + y) / 2, from which it follows that 4-2y = 1 + y.

Grouping the unknown "y" on the same side results in y = 1. Now that "y" is known, we proceed to find the value of "x". Substituting y = 1, we get that x = 2-1 = 1.

Therefore, the common solution between the equations Eq1 and Eq2 is x = 1, y = 1.

Third Exercise

Solve the system of equations Eq1: x + y = 2, Eq2 = 2x-y = 1 using the reduction method.

Solution

The reduction method consists of multiplying the equations given by the appropriate coefficients, so that when these equations are added one of the variables is canceled.

In this particular example, it is not necessary to multiply any equation by any coefficient, just add them. By adding Eq1 plus Eq2, we obtain that 3x = 3, from which we obtain that x = 1.

When evaluating x = 1 in Eq1, we obtain that 1 + y = 2, from which it follows that y = 1.

Therefore, x = 1, y = 1 is the only solution to the simultaneous equations Eq1 and Eq2.

Fourth Exercise

Solve the system of simultaneous equations Eq1: 2x-3y = 8 and Eq2: 4x-3y = 12.

Solution

In this exercise, no particular method is required, therefore the method that is most comfortable for each reader can be applied.

In this case, the reduction method will be used. Multiplying Eq1 by -2 gives the equation Eq3: -4x + 6y = -16. Now, adding Eq3 and Eq2 we get that 3y = -4, therefore y = -4 / 3.

Now, when evaluating y = -4 / 3 in Eq1, we obtain that 2x-3 (-4/3) = 8, from where 2x + 4 = 8, therefore, x = 2.

In conclusion, the only solution to the system of simultaneous equations Eq1 and Eq2 is x = 2, y = -4 / 3.

Observation

The methods described in this article can be applied to systems with more than two simultaneous equations.

The more equations and the more unknowns there are, the more complicated the procedure to solve the system is.

Any method of solving systems of equations will yield the same solutions, that is, the solutions do not depend on the method applied.

References

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