Question

The difference between two numbers is 8. If 12 is added to the bigger number the result will be three times the smaller number. Find the numbers.

Answer 1

Hint-In this question, we use the substitution method to solve the linear equation in two variables. In this method, the value of one variable from one equation is substituted in the other equation. In this way, a pair of the linear equations gets transformed into one linear equation with only one variable, which can then easily be solved.
Complete step-by-step solution -
Let the two numbers be x and y, where x is greater than y.
Now, the difference between two numbers is 8 and we know x is greater than y so, $x - y = 8.............\left( 1 \right)$
Now the 12 is added to the greater number (x) so the result will be three times the smaller number (y).
$
   \Rightarrow x + 12 = 3 \times y \\
   \Rightarrow x + 12 = 3y.............\left( 2 \right) \\
$
We have two equations (1) and (2). So, we use the substitution method to solve the linear equation in two variables.
From (1) equation,
$ \Rightarrow x = 8 + y............\left( 3 \right)$
Now, substitute the value of x in (2) equation.
$
   \Rightarrow 8 + y + 12 = 3y \\
   \Rightarrow 3y - y = 12 + 8 \\
   \Rightarrow 2y = 20 \\
   \Rightarrow y = \dfrac{{20}}{2} \\
   \Rightarrow y = 10 \\
$
Now put the value of y in (3) equation.
$
   \Rightarrow x = 8 + 10 \\
   \Rightarrow x = 18 \\
$
Therefore the numbers are 18 and 10.

Note-In such types of problems we have to solve the linear equations in two variables so we can solve these equations in two ways first by substitution method (mention in above) and second way is elimination method in this method we either add or subtract the equations to get an equation in one variable. When the coefficients of one variable are opposites we add the equations to eliminate a variable and when the coefficients of one variable are equal we subtract the equations to eliminate a variable.