Question Video: Writing and Solving a System of Linear Equations in Three Unknowns Mathematics

Video Transcript

The sum of the ages of three brothers is 123 years. The first brother is three years older than the second brother who is nine years older than the third. Find their current ages.

Well, firstly, it’s important to realize we’re not going to use a trial and error. We’re going to find a way to express the ages of each of the three brothers using algebra. Let’s call the first brother 𝐴. We’ll say that brother 𝐴 is 𝑥 years old. We’ll call the second brother 𝐵, and he’s 𝑦 years old. And finally, the third brother we’ll call him brother 𝐶, and he’s 𝑧 years old. The sum of the ages of the three brothers is 123 years. So we can say that 𝑥 plus 𝑦 plus 𝑧 must be equal to 123.

Secondly, we know that the first brother who we called brother 𝐴 is three years older than the second brother; we called him brother 𝐵. So we can say that 𝑥 must be equal to 𝑦 plus three. Finally, we’re told that brother 𝐵 is nine years older than the third brother who we called 𝐶. So 𝑦 is equal to 𝑧 plus nine is our third equation. Now, we can actually create a third equation in 𝑧. We’re going to replace 𝑦 in the equation 𝑥 equals 𝑦 plus three with 𝑧 plus nine. And we find that 𝑥 is equal to 𝑧 plus nine plus three which is equal to 𝑧 plus 12. So we have an equation in terms of 𝑥, 𝑦, and 𝑧; one in terms of 𝑥 and 𝑧; and one simply in terms of 𝑦 and 𝑧.

We can replace 𝑥 in our first equation with 𝑧 plus 12 and 𝑦 with 𝑧 plus nine. That will leave us an equation simply in terms of 𝑧. When we do, we obtain 𝑧 plus 12 plus 𝑧 plus nine plus 𝑧 equals 123. Now, of course, addition is commutative. It can be done in any order. So we don’t need the parentheses. And what we can do is collect like terms. We have one, two, three 𝑧. And we also find that 12 plus nine is equal to 21. So we have an equation in terms of one single variable. Three 𝑧 plus 21 equals 123. We can solve this equation for 𝑧 by subtracting 21 from both sides. And we find that three 𝑧 is equal to 102. Finally, we divide through by three. And we find 𝑧 is equal to 34. So brother 𝐶 is 34 years old.

We now have everything we need to calculate the ages of the other two brothers. If we go back to two of our earlier equations, these were 𝑥 equals 𝑧 plus 12 and 𝑦 equals 𝑧 plus nine. We can now replace 𝑧 with 34 in each of these equations. When we do, we find that 𝑥 is equal to 34 plus 12 which is equal to 46. And 𝑦 is equal to 34 plus nine which is equal to 43. In ascending order, the ages of the three brothers are 34 years, 43 years, and 46 years.

Now, it’s important to realize we can also check what we’ve done. Let’s go back to our original equation that says that 𝑥 plus 𝑦 plus 𝑧 equals 123. We’ll replace 𝑥 with 46, 𝑦 with 43, and 𝑧 with 34. And we’re hoping that the statement 46 plus 43 plus 34 equals 123 is true. Well, in fact, it is. It is indeed equal to 123, which tells us we must have done are working out correctly. The ages are 34, 43 years, and 46 years.