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The sum of the ages of three brothers is 123 years.
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The first brother is three years older than the second brother who is nine years older than the third.
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Find their current ages.
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Well, firstly, itβs important to realize weβre not going to use a trial and error.
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Weβre going to find a way to express the ages of each of the three brothers using algebra.
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Letβs call the first brother π΄.
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Weβll say that brother π΄ is π₯ years old.
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Weβll call the second brother π΅, and heβs π¦ years old.
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And finally, the third brother weβll call him brother πΆ, and heβs π§ years old.
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The sum of the ages of the three brothers is 123 years.
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So we can say that π₯ plus π¦ plus π§ must be equal to 123.
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Secondly, we know that the first brother who we called brother π΄ is three years older than the second brother; we called him brother π΅.
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So we can say that π₯ must be equal to π¦ plus three.
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Finally, weβre told that brother π΅ is nine years older than the third brother who we called πΆ.
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So π¦ is equal to π§ plus nine is our third equation.
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Now, we can actually create a third equation in π§.
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Weβre going to replace π¦ in the equation π₯ equals π¦ plus three with π§ plus nine.
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And we find that π₯ is equal to π§ plus nine plus three which is equal to π§ plus 12.
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So we have an equation in terms of π₯, π¦, and π§; one in terms of π₯ and π§; and one simply in terms of π¦ and π§.
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We can replace π₯ in our first equation with π§ plus 12 and π¦ with π§ plus nine.
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That will leave us an equation simply in terms of π§.
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When we do, we obtain π§ plus 12 plus π§ plus nine plus π§ equals 123.
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Now, of course, addition is commutative.
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It can be done in any order.
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So we donβt need the parentheses.
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And what we can do is collect like terms.
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We have one, two, three π§.
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And we also find that 12 plus nine is equal to 21.
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So we have an equation in terms of one single variable.
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Three π§ plus 21 equals 123.
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We can solve this equation for π§ by subtracting 21 from both sides.
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And we find that three π§ is equal to 102.
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Finally, we divide through by three.
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And we find π§ is equal to 34.
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So brother πΆ is 34 years old.
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We now have everything we need to calculate the ages of the other two brothers.
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If we go back to two of our earlier equations, these were π₯ equals π§ plus 12 and π¦ equals π§ plus nine.
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We can now replace π§ with 34 in each of these equations.
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When we do, we find that π₯ is equal to 34 plus 12 which is equal to 46.
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And π¦ is equal to 34 plus nine which is equal to 43.
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In ascending order, the ages of the three brothers are 34 years, 43 years, and 46 years.
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Now, itβs important to realize we can also check what weβve done.
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Letβs go back to our original equation that says that π₯ plus π¦ plus π§ equals 123.
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Weβll replace π₯ with 46, π¦ with 43, and π§ with 34.
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And weβre hoping that the statement 46 plus 43 plus 34 equals 123 is true.
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Well, in fact, it is.
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It is indeed equal to 123, which tells us we must have done are working out correctly.
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The ages are 34, 43 years, and 46 years.