8
ADDING LIKE TERMS
LIKE TERMS have the same literal factor (or factors).
2x + 3y + 4x − 5y.
The like terms are 2x and 4x, 3y and −5y.
What do we do with like terms? We add them:
2x + 3y + 4x − 5y = 6x − 2y.
That is, we add their coefficients. (Lesson 3.) The coefficient of a literal factor is its numerical factor.
Actually, the coefficient of any factor is all the remaining factors. Thus in the term 2xy, the coefficient of x is 2y. In this term, x(x − 1), the coefficient of (x − 1) is x.
Problem 1 . 2x + 3y + z
a) What number is the coefficient of x?
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b) What number is the coefficient of y ? 3
c) What number is the coefficient of z ? 1. z = 1· z
Problem 2 . 5x − 4y − z
a) What number is the coefficient of x ? 5
b) What number is the coefficient of y ? −4. We include the minus sign. Lesson 3.
c) What number is the coefficient of z ? −1. −z = (−1)z
Problem 3. What number is the coefficient of x?
| a) | x 2 |
. | 1 2 |
. | Compare Lesson 5, Problem 7b. |
| b) | 3x 4 |
. | 3 4 |
. | 3x 4 |
= | 3 4 |
· x | Lesson 4. |
Problem 4. How do we add like terms?
Add their coefficients; make that sum the coefficient of the literal factor(s).
Problem 5. Add like terms.
| a) 6x + 2x | = 8x | b) 6x − 2x | = 4x | |
| c) 5x + x | = 6x | d) 5x − x | = 4x | |
| e) −4x + 5x | = x | f) 4x − 5x | = −x | |
| Typically we do not write the coefficients 1 or −1. | ||||
| g) −5x − 3x | = −8x | h) −x − x | = −2x | |
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
Problem 6. Add like terms.
a) 2a + 3b These are not like terms. The literals are different.
b) 2a + 3b + 4a − 5ab
= 6a + 3b − 5ab.
Terms that we cannot combine, we simply rewrite.
Problem 7. Remove parentheses and add like terms.
| a) (2a − 3b + c) + (5a − 6b + c) | = | 2a − 3b + c + 5a − 6b + c |
| = | 7a − 9b + 2c | |
| b) (a + 2b + 4c − 3d) − (3a − 8b − 2c + d) | |
| = a + 2b + 4c − 3d −3a + 8b + 2c − d | |
| = −2a + 10b + 6c − 4d | |
| c) (4x − 3y) + (3y − 5x) + (5z − 4x) | |
| = 4x − 3y + 3y − 5x + 5z − 4x | |
| = −5x + 5z | |
| d) (5xy − 3x + 2y − 1) − (2xy − 7x − 8y + 6) | |
| = 5xy − 3x + 2y − 1 −2xy + 7x + 8y − 6 | |
| = 3xy + 4x + 10y − 7 | |
| e) (x − y) − (y + xy − x) − (2x − 4xy − 2y) | |
| = x − y −y − xy + x − 2x + 4xy + 2y | |
| = 3xy | |
| f) (4x² − 7x − 3) − (x² − 4x + 1) | |
| = 4x² − 7x − 3 − x² + 4x − 1 | |
| = 3x² − 3x − 4 | |
| g) (6x3 + 4x² − 2x − 6) − (2x3 − 8x² + x − 2) | |
| = 6x3 + 4x² − 2x − 6 − 2x3 + 8x² − x + 2 | |
| = 4x3 + 12x² − 3x − 4 | |
| h) (x² + x + 1) + (2x² + 2x + 2) − (x² − x − 1) | |
| = x² + x + 1 + 2x² + 2x + 2 − x² + x + 1 | |
| = 2x² + 4x + 4 | |
Problem 8. 5abc + 2cba. Are these like terms?
Yes. The order of factors does not matter.
Upon adding those like terms, we get 7abc.
When writing the final answer, it is conventional to preserve the alphabetical order.
Problem 9. Add like terms.
| a) | 4xy − 9yx = −5xy | b) | 8x − 5xy − 4x + 4yx = 4x − xy |
c) 9xyz + 3yzx + 5zxy = 17xyz
d) 3xy − 4xyz + 3x − 8yx + 5yzx − 9x = −5xy + xyz − 6x
Problem 10. Add like terms.
a) n + (n + 1) = n + n + 1 = 2n + 1
b) n − (n − 1) = n − n + 1 = 1
c) (2n + 1) − (n − 1) = 2n + 1 − n + 1 = n + 2
The subtrahend
"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.
Example. Subtract 2x − 3 from 5x − 4
Solution. 2x − 3 is the subtrahend.
| (5x − 4) − (2x − 3) | = | 5x − 4 − 2x + 3 |
| = | 3x − 1. | |
Notice: The signs of the subtrahend change.
2x − 3 changes to −2x + 3.
And then we add.
Problem 11. Subtract 4a − 2b from a + 3b.
Change the signs of the subtrahend, and add:
a + 3b − 4a + 2b = −3a + 5b
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