Question

The average of two numbers is $7$.Find the numbers if three times one of the numbers is one half of the other number.

Short answer

The required numbers are $2$and $12$.

Step-by-step solution

Step-1 – Form the equations 

Let us suppose the two numbers be x and y.

Now according to the question, the average of both the number is $7$.

The equation so formed is below.

$\begin{array}{c}\frac{x+y}{2}=7\\ x+y=7\left(2\right)\\ x+y=14\end{array}$

Consider the second statement.

Three times one of the numbers that is 3x is one half of the other that means $\frac{1}{2}y$which is mathematically expressed in the equation as shown.

$\begin{array}{c}3x=\frac{1}{2}y\\ 3x\left(2\right)=y\\ 6x=y\\ y=6x\end{array}$

So, both the equations so formed for the given question are

$\begin{array}{c}x+y=14\\ y=6x\end{array}$

Step-2 – Apply the substitution method of solving equations

The algebraic method of substitution involves solving the one of the two equations for one variable in terms of other variable and then substituting the expression so formed for the variable in the second equation.

Step-3 – Substitute the expression 

Since, yis already expressed in terms of xso substitute $y=6x$in the equation first and solve as shown below.

$\begin{array}{c}x+y=14\\ x+6x=14\\ 7x=14\\ x=2\end{array}$

Thus, the value of x is $2$.

Step-4 – Substitute the value of variable

To find the value of y, substitute $x=2$in the equation $x+y=14$and then solve as shown.

$\begin{array}{c}x+y=14\\ 2+y=14\\ y=14-2\\ y=12\end{array}$

Thus, the value of y is $12$.

Hence, the numbers are $2$ and $12$.