Cheatsheet: Linear Equations

Theory

This topic lays the foundation for other topics and hence it is extremely important to know how to solve linear equations. Apart from direct questions from linear equations, many questions on other topics also involve solving linear equations from time to time. It is important not to make silly errors in this topic as it is one of the easiest topics in quantitative aptitude.

Solving linear equations:

• Aggregate the constant terms and variable terms.
• One method to solve two linear equations of two variables is to use one equation to find the value of one variable in terms of the other variable and then substitute this value in the other equation. This will get you the solution for one variable. Now, substitute this value in any of the equations to get the solution of the other variable.
• Another method to solve the equations ax + by = c and mx + ny = p is to multiply each equation with a constant to make either the x-coefficients or the y-coefficients of both the equations equal and then subtract the equations to get an equation in one variable.
  

Tips 

The solutions of two equations ax + by = c and mx + ny = p depend on the following conditions: 

• If a/m = b/n ≠ c/p , then the slopes of both the lines are equal but the ratio of intercepts is different and hence the lines are parallel, which implies that there is no point of intersection. 
• a/m ≠  b/n, then the slopes are not equal and hence the lines must intersect at a point, which gives us a unique solution. 
• If a/m = b/n = c/p , then the lines have equal slopes and equal intercepts. Hence, they are the same lines and as they have infinite points in common, the number of solutions is infinite.

Solved Example