Algebra Article 53 Exercises
by William C. Michael
Algebra is a necessary study for modern students seeking a high school diploma and/or college admission. Students must work to master this study — and there is no excuse why any classical educated student cannot do so. It is an easy study that requires effort like any other study.
Let us consider the simple exercises required by the assignment for Article 53 in Algebra I.
Demonstrate your understanding of the lesson by explaining, step-by-step, the solution of the following exercises.1. What is the sum of ab, 3ab, 6ab, and 10ab?2. What is the sum of 8x-4y, 5x-3y, 7x-6y, and 6x-2y?
First, note that the assignment requires you to “demonstrate your understanding”. We are not seeking the correct answer. We are seeking a demonstration that you understand the meaning of these expressions and the principles of addition in Algebra.
Second, note that the assignment asks for a “step-by-step” explanation. That means you should explain how you reach the solution one step at a time.
1. What is the sum of ab, 3ab, 6ab, and 10ab?
This question tests your understanding of the meaning of algebraic expressions, and the addition of such. If you have studied previous lessons honestly and carefully, this should be easy. If you have lazily fudged your way through previous lessons, you’ll now find yourself stuck.
You have learned the rules for adding quantities that are similar, which have similar signs. You should demonstrate that you know the rule that applies to the problem:
“Add together the coefficients of the several quantities; to their sum prefix the common sign, and annex the common letter or letters.”
So, first, we will add the coefficients of the quantities.
1 + 3 + 6 + 10 = 20
Then, we will annex the common letters, to make 20ab.
But how do we know this is true? Do we just do what the rule says, mindlessly? No.
We understand that multiplication is merely a quick way to add a number of items. 3ab is simply a short way of writing 3 ab’s, or ab + ab + ab. If we add to this another ab, we would have:
ab + ab + ab + ab
which makes 4 ab’s, or 4ab.
Therefore, the rule that tells us to add the coefficients in 3ab + 1ab, is certainly true.
We do the same when we add 6ab (i.e., 6 more ab’s) and 10ab (10 more ab’s).
4 ab’s + 6 ab’s make 10 ab’s; and 10 more ab’s make 20ab’s, or 20ab, which is the correct sum.
Now, we can understand why the rule to add the coefficients is true.
2. What is the sum of 8x-4y, 5x-3y, 7x-6y, and 6x-2y?
This problem is more complicated than the first question. We have two different quantities being added and subtracted. One is represented by the letter x, and another by the letter y. We cannot add or subtract x and y as in Arithmetic, so what can we do?
First, let’s think about what we’re doing.
We are seeking to find the sum of x’s and the sum of y’s.
Let us, then, look first at the quantity x. How many do we have?
8x + 5x + 7x + 6x.
Following our rule, we add the coefficients 8 + 5 + 7 + 6, which makes 26. To this we annex the common letter, to make 26 x’s, or 26x.
Then, we add the second quantity y. How many do we have?
-4y + -3y + -6y+ -2y.
Following our rule, we add the coefficients -4 + -3 + -6+ -2, which makes -15. To this we annex the common letter, to make -15 y’s, or -15y.
Since we have two different quantities, we cannot add them, so we simply write the sum that we have:
26x + -15y, which is equal to 26x — 15y.
Algebra is a challenging subject that is necessary for modern students. There is no reason why students enjoying a classical Catholic education should not master the subject. By carefully studying each point in our Algebra course, and seeing to demonstrate our understanding, we can enjoy and master Algebra.