Simultaneous Linear Equations

Simultaneous equations are two or more equations, with the same unknowns (variables) and solutions. They are solved by using one of two methods: Elimination or Substitution. Below are examples of simultaneous equations.

Elimination

In this method, the first objective is to eliminate one of the two unknowns (variables). This is done by:

1.  Adding the equations. This procedure is carried out if the coefficients of one of the unknowns are the same, but they have different signs.

2. Subtracting the equations. This is done if the coefficients of one of the unknowns are the same and have the same sign.

3. Multiply one or both equations by a number(s) then add or subtract. If the coefficients of one of the unknowns are not the same, multiply one or both equation by a number(s) which will make the coefficients of one of the unknowns the same. Then, add or subtract the equations (depending on if they satisfy #1 or #2 above).

Having eliminated one of the unknowns, solve for the value of the remaining unknown. On finding its value, substitute it in one of the two equations, and solve for the value of the remaining unknown.

Example

Note:  Re-visit the rules in carrying numbers across the equal sign in Linear Equations (with one unknown), if needs be.

Substitution

To use the method of substitution when solving simultaneous equations, follow the steps listed below, illustrated using the simultaneous equations:

Step 1: Using one of the equations, make one of the unknowns the subject of that equation (that is, have it alone on one side of the equal sign).

Step 2: Substitute the value of the subject in the other equation and solve for the value of the remaining unknown.

Step 3: Solve for the unknown which was made the subject, by substituting the value obtained in step 2 in the equation made by the subject.

Example