Download Activity 8 - Mathematics

ACTIVITY 8:
Basic Equations (Section 1.1, pp. 74-80)
An equation is a statement that two mathematical
expressions are equal. For instance,
43 − 2 * 42 = 32.
Most equations that we study in Algebra contain
variables. For example,
x3 − 2x2 = 32.
Given an equation in the variable (let’s say) x, the goal is
to find the values of x that make the equation true;
these values are called the solutions or roots of the
equation, and the process of finding the solutions is
called solving the equation.
Example 2: Determine whether the given value of
x is a solution of the equation:
1 - [2 - (3 - x)]  4x - (6  x)
x=4
LHS  1  2  3  4  1  2  1  1  2  1  1 3  2
RHS  4 * 4  6  4  16  10  6
Because 6 and -2 are different
numbers x = 4 is NOT a solution to the
above equation.
Example 3: The given equations are linear or equivalent
to linear equations. Solve these equations:
2
1
y 1
y   y  3 
3
2
4
1
2
  y  1
12  y   y  3  
12

2
3
  4 
LCD = 12
12 2
12 1
12 y  1
* y  *  y  3  *
1 3
1 2
1
4
14 y  18  3 y  3
4 * 2 y  6 y  3  3 y  1
 3 y  18  3 y  18
8 y  6 y  18  3 y  3
11 y  21
21
y
11
(x  3)2  (x - 1)2 - 8
x  3x  3  x - 1x - 1 - 8
x 2  3x  3x  9  x 2  x  x  1 - 8
x2  6x  9  x2  2 x  7
 x  2 x  9 x  2 x  9
2
2
8x  16
 16
x
 2
8
1
5
2
 2

x3 x 9 x3
1
5
2


x  3 x  3x  3 x  3
LCD = (x+3)(x-3)
 1

5
LHS  x  3x  3


 x  3 x  3x  3 
1
5
 x  3x  3
 x  3x  3
x  3x  3
x3
 x  3  5
 x2
x  2  2x  6
x 6 x 6
2
 x  32  2 x  6
RHS   x  3 x  3
x3
4  x
Solving a Simple nth Degree
Equation
The real solutions of the equation xn = a are:
•x=
n
•x=±
a , if n is odd
n
a , if n is even and a ≥ 0
• no real solutions, if n is even and a < 0
x  49
2
x   49
x7
3x - 5 - 15  0
2
x 2  16  0
x 2  - 16
NO SOLUTION!
x - 5
2
5
3x - 5  15
x -5   5
3x - 5 15

3
3
x  5 5
2
2
y 5
3
1
3
y 5

y


1
3
3

  53


y  53
y  125
x 5  32
x  5 32
x2
Example 5:
The average daily food consumption F of an herbivorous mammal with body
weight w, where both F and w are measured in pounds, is given
approximatively by the equation
F = 0.3w3/4.
Find the weight w of an elephant who consumes 300 lb of food per day.
300  .3w
3
300
 w  4
.3
1000  w
3
4
3
4
1000
10 
4
3
4
3 3
w
w
10 4  w
Example 6: Solve the given
equation for the indicated variable.
mM
F G 2 ;
r
LCD =
r2
For m
Fr 2  GmM
Fr 2 GmM

GM
GM
Fr 2
m
GM
a 1 a 1 b 1


; For a.
b
b
a
LCD = ab
 a  1   aa  1
LHS  ab

 b 
 a 1 b 1 
 a 1 
 b  1   aa 1  bb  1
RHS  ab

  ab

  ab
a 
 b 
 a 
 b
 a2  a  b2  b
a2  a  a2  a  b2  b
 a2
 a2
a  a  b 2  b
a a
2a  b 2  b
b2  b
a
2