Simultaneous Equations

A simultaneous equation is formed by two or more equations with two or more unknowns and a common solution.

The solution to the system is a pair of numbers x_1, y₁, such that replacing x by x₁ and y by y₁, both equations are verified.

x = 2, y = 3

Equivalent Simultaneous Equations

1. If both members of an equation in simultaneous equations are added or subtracted by the same expression, the result is an equivalent simultaneous equation.

x = 2, y = 3

2.If both members of an equation in a simultaneous equation are multiplied or divided by a nonzero number, the resultant is an equivalent simultaneous equation.

x = 2, y = 3

3. If an equation of a system is added or subtracted by another equation of the same system, the resultant system is equivalent.

x = 2, y = 3

4.If in a system, an equation is replaced by another equation that is obtained from adding the two equations from a system previously multiplied or divided by a nonzero number, the resultant system is equivalent.

5. If the order of the equations or the order of the unknowns changes, it is another equivalent system.

Generally, two methods are used to solve simultaneous equations:

1. Simultaneous Equations by Substitution

1. Isolate an unknown in one of the equations.

2. Substitute the expression of this unknown into the other equation in order to obtain an equation with only one unknown.

3. Solve the equation.

4. The value obtained is substituted into the other equation.

5. The two values obtained are the solutions to the system.

1Isolate x.

2Substitute it into the other equation.

3 Solve the equation.

4Substitute the value.

5Solution:

2. Simultaneous Equations by Elimination

1.Adapt the two equations, multiplying by the appropriate numbers in order to eliminate one of the unknown values.

2.Add them, and eliminate one of the unknowns.

3.Solve the resulting equation.

4.Substitute the value obtained in one of the initial equations and then solve.

5.The two values obtained are the solution of the system.

The easiest method is to remove the y, this way the equations do not have to be prepared. However, by choosing to supress the x, the process can be seen much better.

Add and solve the equation:

Replace the value of y in the second equation.

Solution: