Question

Three times one number added to five times another number is $54$. The second number is two less than the first. Find the numbers.

Short answer

The required numbers are $8$and $6$.

Step-by-step solution

Step-1 – Form the equations 

Let us suppose the two numbers be x and y.

Now according to the question, three times the one number means 3x and five times of another number that is 5y, and the addition of both the numbers results in $54$.

The equation so formed is below.

$3x+5y=54$

Consider the second statement.

The second number that is y is two less than the first that is x which is mathematically expressed in the equation as shown.

$y=x-2$

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

$\begin{array}{c}3x+5y=54\\ y=x-2\end{array}$

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 $y=x-2$in the equation first and solve as shown below.

$\begin{array}{c}3x+5y=54\\ 3x+5\left(x-2\right)=54\\ 3x+5x-10=54\\ 8x=64\end{array}$

Now divide both sides by 8 and simplify it further as.

$\begin{array}{c}8x=64\\ x=8\end{array}$

Thus, the value of x is $8$.

Step-4 – Substitute the value of variable

To find the value of y, substitute $x=8$in the equation $y=x-2$and then solve as shown.

$\begin{array}{c}y=x-2\\ y=8-2\\ y=6\end{array}$

Thus, the value of y is $6$.

Hence, the numbers are $8$ and $6$.