## 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.

Theory

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.

Tip

If you have n equations to solve for n+1 variables, then the solution is indeterminate.

Tip

If you have n equations, along with some additional conditions on variables, to solve for n+1 variables, then the solution may be determinate.

Tip

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

• If $$\frac{a}{m} = \frac{b}{n} \neq \frac{c}{p}$$, 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.

• $$\frac{a}{m} \neq \frac{b}{n}$$, then the slopes are not equal and hence the lines must intersect at a point, which gives us a unique solution.

• If $$\frac{a}{m} = \frac{b}{n} = \frac{c}{p}$$, 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

Solve the equations x + 3y = 12 and 3x + 5y = 20.

Explanation:
Using the first equation, you get x = 12 – 3y. Now, by substituting this in the second equation we get 3(12 – 3y) + 5y = 20 => 4y = 16 => y = 4. Now, by substituting the value of y in the first equation, we get x + 3(4) = 12 => x = 0.

Solved Example

Solve the equations 2x + 3y = 12 and 3x + 5y = 19.

Explanation:
In order to make the x-coefficients equal, multiply the first equation and the second equation with 3 and 2 respectively. You get 6x + 9y = 36 and 6x + 10y = 38. Now, if you subtract these two, you get the y = 2. By substituting the value of y in any of the equations, you get x = 3.