ADDING LIKE TERMS
5x + 3x − 2x.
How many x 's are there? 5 + 3 − 2 equals 6 of them. We write:
5x + 3x − 2x = 6x.
We call 5, 3, and −2 the coefficients respectively of each term. Recall that in naming terms we include the minus sign.
Here, on the other hand, is a sum of unlike terms:
x² − 2xy + y²
There is no way that we could combine them.
The most elementary terms that we could combine are a and −a.
5 + (−5) = 0.
Adding like terms
In this sum—
2x − 3y + 4x + 5y
—the like terms are 2x and 4x, y and −5y.
Upon combining, or adding, them:
2x + 3y + 4x − 5y = 6x − 2y.
That has the effect of reducing the number of terms. Which is what we like.
We say that there are two terms "in" x and two "in" y. The preposition "in" indicates which are the like terms.
See Problem 12.
Problem 1 . 6a − 3b + c − d.
a) What number is the coefficient of a ?
To see the answer, pass your mouse over the colored area.
b) What number is the coefficient of b ? −3
c) What number is the coefficient of c ? 1. For c = 1·c.
d) What number is the coefficient of d ? −1. −d = −1·d.
See Lesson 5.
It is the style in algebra not to write the coefficients 1 or −1.
Actually, the coefficient of any factor is all the remaining factors. Thus in the term 4ab, the coefficient of a is 4b; the coefficient of 4a is b; and so on. In this term, x(x − 1), the coefficient of (x − 1) is x.
Problem 2. In the expression 5ayx, name the coefficient of
a) x 5ay b) y 5ax c) yx 5a
d) 5a xy e) 5 ayx
Problem 3. In this product 2(x + y)z
a) name each factor. 2, (x + y), z
b) name the coefficient of z. 2(x + y)
c) name the coefficient of (x + y). 2z
Problem 4. What number is the coefficient of x?
Problem 5. How do we add like terms?
Add their coefficients.
Problem 6. Add like terms.
i) −3x − 4 + 2x + 6 = −x + 2
j) x − 2 − 4x − 5 = −3x − 7
k) 4x + y − 2x + y = 2x + 2y
l) 3x − y − 8x + 2y = −5x + y
m) 4x² − 5x² + x² = 0
Problem 7. Add like terms.
a) 2a + 3b These are not like terms. The literals are different.
b) 2a + 3b + 4a − 5ab
= 6a + 3b − 5ab.
Problem 8. Remove parentheses and add like terms.
Problem 9. 5abc + 2cba. Are those like terms?
Yes. The order of factors does not matter.
Problem 10. Add like terms.
c) 9xyz + 3yzx + 5zxy = 17xyz
d) 3xy − 4xyz + 3x − 8yx + 5yzx − 9x = −5xy + xyz − 6x
Problem 11. Add like terms.
a) 2n + 2 − n = n + 2
b) n − 2 − 3n + 1 = −2n − 1
c) 2n + 4 − 2n − 2 = 2
Problem 12. Add like terms, which are in (x + 2). Do not remove parentheses.
a) 3(x + 2) + 7(x + 2) = 10(x + 2).
b) 2(x + 2) − 5(x + 2) = −3(x + 2).
c) x(x + 2) + 4(x + 2) = (x + 4)(x + 2).
We added the coefficients.
d) x(x + 2) − (x + 2) = (x − 1)(x + 2).
Problem 13. Add like terms, which are in x or y. Add the coefficients.
a) px + qx = (p + q)x.
b) ax + by − cx + dy = (a − c)x + (b +d)y.
c) x + ax = (1 + a)x. d) ax − x = (a − 1)x.
e) (a + b)x + cx = (a + b + c)x.
f) (a − b)x − cx = (a − b − c)x.
f) (a + b)x − (b + a)x = 0.
Problem 14. Add like terms.
a) 3a2b3 − 2ab2 + a3b2 − 5b2a + b3a2 = 4a2b3 − 7ab2 + a3b2.
b) xy2 − xy + x2y − y2x + 2yx2 + yx = 3x2y.
In calculus, the student will not see any problem in the form "Subtract a from b." However, in certain standard exams that form tends to come up. Hence, the following rule.
The rule for subtraction
"Subtract a from b." Is that a − b or b − a ?
It is b − a. a is the number being subtracted. It is called the subtrahend. The subtrahend appears to the right of the minus sign -- before the word "from."
Example. Subtract 2x − 3 from 5x − 4
Solution. 2x − 3 is the subtrahend.
Notice: The signs of the subtrahend change.
2x − 3 changes to −2x + 3.
We can therefore state the following rule for subtraction.
Change the signs of all the terms in the subtrahend.
Problem 15. Subtract 4a − 2b from a + 3b.
Change the signs of the subtrahend, and add:
a + 3b − 4a + 2b = −3a + 5b.
Problem 16. Subtract x² − 5x + 7 from 3x² − 8x − 2.
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