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1.05 Variables
and Substitution from Basic Algebra: One Step at a Time © 2002 P. 27-30 Dr. Robert J. Rapalje Seminole State College of Florida
ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE END OF THIS PAGE
The
study of algebra usually begins with systems of numbers and operations
with those numbers. True to
form, in this book you began with number systems, signed numbers, and
order of operations. But
algebra is much more than just the study of numbers and computations.
As you progress in algebra, you will notice that the concepts and
problem solving methods become more complicated and also more abstract.
You will go from concrete computations of numbers that you can see
with your eyes and do with a calculator, to abstract ideas that you
“see” and understand with your mind. The
first step towards abstract thinking in algebra is the introduction of variables.
Up to this point, all problems have been completely numerical.
Variables are simply
letters that take the place of numbers.
Just as a pronoun is a
word that takes the place of a noun,
a variable could be thought of as a “pronumber,”
a letter that takes the place of a number.
You can find the value of an algebraic expression by
substitution if you are given the numbers to be substituted.
Example
1: Find the
value of 3x + 4,
a)
if x = 2;
b) if x = -2;
c) if x = -12 Solution:
Substitute
the given values for x (plug it in, plug it in!)
a)
3(2) + 4
b) 3(-2) + 4
c) 3(-12) + 4
6
+ 4
-6 + 4
-36 + 4
10
-2
-32 At
this point you may be
wondering, “Why x?” Well,
why not x! As the next examples illustrate, any letter(s) can and will
be used. Historically, the
favorite letter used as a variable in math is the letter x.
In fact, usually we use the letters x,
y, and z
to represent variables (quantities that vary or change) and we use the
letters a, b, and c to represent constants (quantities that remain the
same.) Example
2: Find the
value of 3x + 4y - 8, a)
if x = 2 and y = 3;
b) if x = -2 and y = -6;
c) if x = -12 and y
= 4 Solution:
Substitute
the given values for x and y a) 3(2) + 4(3) – 8
b) 3(-2) + 4(-6) – 8
c) 3(-12) + 4(4) - 8
6
+ 12 -
8
-6 - 24 – 8
-36 + 16 -
8
10
-38
-28 Example
3 : Find the value
of x2
+ 4xy - z2,
a)
if x = 2, y = 3, and z = 4;
b) if x = -2, y = -6, and z = -5 Solution:
Substitute
the given values for x, y, and z.
a)
22
+ 4(2)(3) - 42
b) (-2)2
+ 4(-2)(-6) - (-5)2
4 + 24 - 16 4 + 48 - 25
12
27
EXERCISES:
Find the values of the following expressions for the given values
of the
variables. 1.
Find the value of 5x
+ 12
a)
if x = 2;
b) if x = -2;
c) if x = -12
5(
) + 12
5(
) + ____
____ + ____
_____ + 12
_____ +
____
____ + ____
_____
_____
_____ 2.
Find the value of 7x - 4y
- 10,
a)
if x = 3 and y
= 2;
b) if x = -4 and y = -7;
c) if x = -3 and y = 5
7(
) - 4(
) - 10
7( )
- 4( )
- ____
3. Find the value of x2 - 3yz
- z2,
if x
= 5, y = 2, and z = 3.
( )2
- 3( )(
) - (
)2
4.
Find the value of x2 - 3yz
- z2,
if x = -5, y
= -2, and z = -3
(
)2
- 3( )(
) - ( )2
Example
4: Given that x = -2, y = -3, z
= 4, and w = 1, find the value of
Solution:
In
each of the following exercises, substitute the given values and find the
value of the expressions, given that x
= -2, y = -3, z = 4, and w = -1. 5.
xy +
5z
6. xy -
5z
7. xz -
5y
8.
xz +
5y
=(
)( )
+ 5( )
=( )(
) - 5( )
= _____ + _____
=
_____ - _____
= _____________
= _____________
9.
xz -
5w
10.
xz +
5w
11. x2
+ y2
12. z2
+ w2
= ( )2
+ (
)2
= _____________ 13.
x2 -
y2
14. w2 - z2
15. x2
+ y2
+ z2
16. w2
-
y2 -
z2
Find
the values, given that x = -2, y
= -3, z = 4, and w = -1. 17.
x3
+ y3
18. x3
-
y3
19. x3
-
z3
20.
-z2
+ w2
21. x2 + 3xz
+ z2
22. x2
+ yzw
-
w2 23.
26.
ANSWERS 1.05 p. 28 - 30: 1a) 22; b) 2; c) -48; 2a) 3; b) -10; c) -51; 3. -2; 4. -2; 5. 26; 6. -14; 7. 7; 8. -23; 9. ‑3; 10. -13; 11. 13; 12. 17; 13. -5; 14. -15; 15. 29; 16. -24; 17. -35; 18. 19; 19. ‑72; 20. -15; 21. -4; 22. 15; 23. -1; 24. 5; 25. -2; 26. -4; 27. Undefined; 28. 0.
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