Solving an algebraic exercise using only one variable $x$

Question

This exercise is supposed to be solved using only one though I don't see how would that be possible.

Could someone tell how to solve using only one unknown variable $x$?

Thank you.

Exercise. The sum of two numbers is $108$ and the double of the greater exceeds the triple of the minor in $156$. Find the numbers.

I solved using two variables $x$ and $y$ and specifying that $x>y$.

Answer 1

So let's piece this together:

"The sum of two numbers is $108$" - this gives us $x+y = 108$

"the double of the greater exceeds the triple of the minor in 156"- Without loss of generality, we can assume $x>y$, and thus see $2x - 3y = 156$.

Thus, we have a system of equations, which we seek to find $x$ and $y$:

$$x+y = 108$$ $$2x - 3y = 156$$

You could solve this by whatever method you prefer to do so. Since you seek to use only one variable, notice that the first equation implies $y = 108-x$, yielding

$$x+(108-x)= 108$$ $$2x - 3(108-x) = 156$$

This is a technique known as solving by substitution: solving for one of the variables and then substituting the resulting expression into the other equations.

Notice, too, if you used the second of the equations from our original system, you could have found

$$y = \frac{156-2x}{-3}$$

You also could solve for $x$ instead and use that as your substitution! Using the first equation,

$$x = 108-y$$

or perhaps using our second equation

$$x = \frac{156+3y}{2}$$

Any of these four substitutions is completely valid. These substitutions turn one of the equations into a true statement (notice how our original substitution turned the first equation, after simplifying, into $108=108$?). The other equation then becomes an equation of just one variable (after simplifying, the second equation in our original substitution becomes $-x-324=156$).

This also is a valid method if you have $n$ equations in $n$ variables, too, but it can become more complicated since you have to solve for all but one of the variables and substitute them into an equation. It can get messy pretty fast.

Answer 2

If one of the numbers is $x$ and the sum of the two is 108, the other number is $108 - x$. If $x \ge 108 - x$, we write

$2x = 3(108 - x) + 156; \tag 1$

$2x = 324 - 3x + 156 = 480 - 3x; \tag 2$

$5x = 480; \tag 3$

$x = 96; \tag 4$

$108 - x = 12. \tag 5$

OR, if $108 - x \ge x$,

$2(108 - x) = 3x + 156; \tag 6$

$216 - 2x = 3x + 156; \tag 7$

$5x = 60; \tag 8$

$x = 12; \tag 9$

$108 - x = 96. \tag{10}$