Solution - Adding, subtracting and finding the least common multiple
Step by Step Solution
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
b/7+b/3-(23/9-b/14)=0
Step by step solution :
Step 1 :
b
Simplify ——
14
Equation at the end of step 1 :
b b 23 b
(—+—)-(——-——) = 0
7 3 9 14
Step 2 :
23
Simplify ——
9
Equation at the end of step 2 :
b b 23 b
(— + —) - (—— - ——) = 0
7 3 9 14
Step 3 :
Calculating the Least Common Multiple :
3.1 Find the Least Common Multiple
The left denominator is : 9
The right denominator is : 14
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 3 | 2 | 0 | 2 |
| 2 | 0 | 1 | 1 |
| 7 | 0 | 1 | 1 |
| Product of all Prime Factors | 9 | 14 | 126 |
Least Common Multiple:
126
Calculating Multipliers :
3.2 Calculate multipliers for the two fractions
Denote the Least Common Multiple by L.C.M
Denote the Left Multiplier by Left_M
Denote the Right Multiplier by Right_M
Denote the Left Deniminator by L_Deno
Denote the Right Multiplier by R_Deno
Left_M = L.C.M / L_Deno = 14
Right_M = L.C.M / R_Deno = 9
Making Equivalent Fractions :
3.3 Rewrite the two fractions into equivalent fractions
Two fractions are called equivalent if they have the same numeric value.
For example : 1/2 and 2/4 are equivalent, y/(y+1)2 and (y2+y)/(y+1)3 are equivalent as well.
To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.
L. Mult. • L. Num. 23 • 14 —————————————————— = ——————— L.C.M 126 R. Mult. • R. Num. b • 9 —————————————————— = ————— L.C.M 126
Adding fractions that have a common denominator :
3.4 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
23 • 14 - (b • 9) 322 - 9b
————————————————— = ————————
126 126
Equation at the end of step 3 :
b b (322 - 9b)
(— + —) - —————————— = 0
7 3 126
Step 4 :
b
Simplify —
3
Equation at the end of step 4 :
b b (322 - 9b)
(— + —) - —————————— = 0
7 3 126
Step 5 :
b
Simplify —
7
Equation at the end of step 5 :
b b (322 - 9b)
(— + —) - —————————— = 0
7 3 126
Step 6 :
Calculating the Least Common Multiple :
6.1 Find the Least Common Multiple
The left denominator is : 7
The right denominator is : 3
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 7 | 1 | 0 | 1 |
| 3 | 0 | 1 | 1 |
| Product of all Prime Factors | 7 | 3 | 21 |
Least Common Multiple:
21
Calculating Multipliers :
6.2 Calculate multipliers for the two fractions
Denote the Least Common Multiple by L.C.M
Denote the Left Multiplier by Left_M
Denote the Right Multiplier by Right_M
Denote the Left Deniminator by L_Deno
Denote the Right Multiplier by R_Deno
Left_M = L.C.M / L_Deno = 3
Right_M = L.C.M / R_Deno = 7
Making Equivalent Fractions :
6.3 Rewrite the two fractions into equivalent fractions
L. Mult. • L. Num. b • 3 —————————————————— = ————— L.C.M 21 R. Mult. • R. Num. b • 7 —————————————————— = ————— L.C.M 21
Adding fractions that have a common denominator :
6.4 Adding up the two equivalent fractions
b • 3 + b • 7 10b
————————————— = ———
21 21
Equation at the end of step 6 :
10b (322 - 9b)
——— - —————————— = 0
21 126
Step 7 :
Calculating the Least Common Multiple :
7.1 Find the Least Common Multiple
The left denominator is : 21
The right denominator is : 126
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 3 | 1 | 2 | 2 |
| 7 | 1 | 1 | 1 |
| 2 | 0 | 1 | 1 |
| Product of all Prime Factors | 21 | 126 | 126 |
Least Common Multiple:
126
Calculating Multipliers :
7.2 Calculate multipliers for the two fractions
Denote the Least Common Multiple by L.C.M
Denote the Left Multiplier by Left_M
Denote the Right Multiplier by Right_M
Denote the Left Deniminator by L_Deno
Denote the Right Multiplier by R_Deno
Left_M = L.C.M / L_Deno = 6
Right_M = L.C.M / R_Deno = 1
Making Equivalent Fractions :
7.3 Rewrite the two fractions into equivalent fractions
L. Mult. • L. Num. 10b • 6 —————————————————— = ——————— L.C.M 126 R. Mult. • R. Num. (322-9b) —————————————————— = ———————— L.C.M 126
Adding fractions that have a common denominator :
7.4 Adding up the two equivalent fractions
10b • 6 - ((322-9b)) 69b - 322
———————————————————— = —————————
126 126
Step 8 :
Pulling out like terms :
8.1 Pull out like factors :
69b - 322 = 23 • (3b - 14)
Equation at the end of step 8 :
23 • (3b - 14)
—————————————— = 0
126
Step 9 :
When a fraction equals zero :
9.1 When a fraction equals zero ...Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.
Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.
Here's how:
23•(3b-14)
—————————— • 126 = 0 • 126
126
Now, on the left hand side, the 126 cancels out the denominator, while, on the right hand side, zero times anything is still zero.
The equation now takes the shape :
23 • (3b-14) = 0
Equations which are never true :
9.2 Solve : 23 = 0
This equation has no solution.
A a non-zero constant never equals zero.
Solving a Single Variable Equation :
9.3 Solve : 3b-14 = 0
Add 14 to both sides of the equation :
3b = 14
Divide both sides of the equation by 3:
b = 14/3 = 4.667
One solution was found :
b = 14/3 = 4.667How did we do?
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