Simultaneous Linear Equations

Meaning of Simultaneous Equations

Simultaneous linear equations are two or more linear equations containing the same variables that must all be true (satisfied) at the same time. Solving them means finding the values of the variables that work for every equation.

Methods of Solving Simultaneous Equations

• Elimination method – add or subtract the equations (after adjusting coefficients) to eliminate one variable
• Substitution method – make one variable the subject of one equation, then substitute it into the other equation
• Graphical method – plot both equations on the same graph; the point where the lines cross is the solution

Worked Example (Elimination)

Solve: x + y = 7 and x − y = 3.

Adding the two equations: 2x = 10, so x = 5. Substituting into the first equation: 5 + y = 7, so y = 2. The solution is x = 5, y = 2.

Worked Example (Substitution)

Solve: y = 2x and x + y = 9.

Substitute y = 2x into the second equation: x + 2x = 9, so 3x = 9, giving x = 3. Then y = 2(3) = 6. The solution is x = 3, y = 6.

Checking Solutions

Always substitute the values found back into both original equations to confirm that they satisfy each one.

Summary

Simultaneous linear equations can be solved by elimination, substitution or graphing, and the correct solution must satisfy all the given equations at once.