Simultaneous Equations

Simultaneous 

equations

-Simultaneous equations refer to a set of equations involving multiple variables that must be solved at the same time. In other words, the solution to each equation in the set must satisfy all of the other equations in the set simultaneously.

For example, consider the following set of simultaneous equations:

3x - y = 12

2x + y = 13

To solve these equations, one needs to find the values of x and y that satisfy both equations at the same time. One approach to solve simultaneous equations is to use the method of substitution or elimination.

In the method of substitution, one of the equations is solved for one variable (usually the one with the easiest coefficient), and then the resulting expression is substituted into the other equation. This reduces the number of variables to one, which can be solved to find its value. Then, the value found is substituted back into one of the original equations to find the other variable.

In the method of elimination, one or more variables are eliminated by adding or subtracting the equations in a way that results in a new equation that contains only one variable. Then, the value of the eliminated variable can be found and substituted back into one of the original equations to find the value of the remaining variable.

In our example above, one can use the method of substitution to solve for x or y in one of the equations and substitute it into the other equation. Alternatively, one can use the method of elimination to add or subtract the equations in a way that eliminates one of the variables. Either way, the goal is to find the values of x and y that satisfy both equations at the same time.

3x - y = 12

2x + y = 13

(3x - y) + (2x + y) = 12 + 13

5x = 25

5x/5 = 25/5

x = 5

substituting 5 for x,

(3 x 5) - y = 12

15 - y = 12

15 - y - 15 = 12 - 15

-y = -3

-y/-1 = -3/-1

y = 3

therefore,

x = 5

y = 3