Solution - Adding, subtracting and finding the least common multiple
Other Ways to Solve
Adding, subtracting and finding the least common multipleStep by Step Solution
Step by step solution :
Step 1 :
2
Simplify —
3
Equation at the end of step 1 :
11 2
((7•(y2))-(——•y))-— = 0
3 3
Step 2 :
11
Simplify ——
3
Equation at the end of step 2 :
11 2 ((7 • (y2)) - (—— • y)) - — = 0 3 3Step 3 :
Equation at the end of step 3 :
11y 2
(7y2 - ———) - — = 0
3 3
Step 4 :
Rewriting the whole as an Equivalent Fraction :
4.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using 3 as the denominator :
7y2 7y2 • 3
7y2 = ——— = ———————
1 3
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Adding fractions that have a common denominator :
4.2 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:
7y2 • 3 - (11y) 21y2 - 11y
——————————————— = ——————————
3 3
Equation at the end of step 4 :
(21y2 - 11y) 2
———————————— - — = 0
3 3
Step 5 :
Step 6 :
Pulling out like terms :
6.1 Pull out like factors :
21y2 - 11y = y • (21y - 11)
Adding fractions which have a common denominator :
6.2 Adding fractions which have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
y • (21y-11) - (2) 21y2 - 11y - 2
—————————————————— = ——————————————
3 3
Trying to factor by splitting the middle term
6.3 Factoring 21y2 - 11y - 2
The first term is, 21y2 its coefficient is 21 .
The middle term is, -11y its coefficient is -11 .
The last term, "the constant", is -2
Step-1 : Multiply the coefficient of the first term by the constant 21 • -2 = -42
Step-2 : Find two factors of -42 whose sum equals the coefficient of the middle term, which is -11 .
| -42 | + | 1 | = | -41 | ||
| -21 | + | 2 | = | -19 | ||
| -14 | + | 3 | = | -11 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -14 and 3
21y2 - 14y + 3y - 2
Step-4 : Add up the first 2 terms, pulling out like factors :
7y • (3y-2)
Add up the last 2 terms, pulling out common factors :
1 • (3y-2)
Step-5 : Add up the four terms of step 4 :
(7y+1) • (3y-2)
Which is the desired factorization
Equation at the end of step 6 :
(3y - 2) • (7y + 1)
——————————————————— = 0
3
Step 7 :
When a fraction equals zero :
7.1 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:
(3y-2)•(7y+1)
————————————— • 3 = 0 • 3
3
Now, on the left hand side, the 3 cancels out the denominator, while, on the right hand side, zero times anything is still zero.
The equation now takes the shape :
(3y-2) • (7y+1) = 0
Theory - Roots of a product :
7.2 A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.
Solving a Single Variable Equation :
7.3 Solve : 3y-2 = 0
Add 2 to both sides of the equation :
3y = 2
Divide both sides of the equation by 3:
y = 2/3 = 0.667
Solving a Single Variable Equation :
7.4 Solve : 7y+1 = 0
Subtract 1 from both sides of the equation :
7y = -1
Divide both sides of the equation by 7:
y = -1/7 = -0.143
Supplement : Solving Quadratic Equation Directly
Solving 21y2-11y-2 = 0 directly Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula
Parabola, Finding the Vertex :
8.1 Find the Vertex of t = 21y2-11y-2
Parabolas have a highest or a lowest point called the Vertex . Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) . We know this even before plotting "t" because the coefficient of the first term, 21 , is positive (greater than zero).
Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.
Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.
For any parabola,Ay2+By+C,the y -coordinate of the vertex is given by -B/(2A) . In our case the y coordinate is 0.2619
Plugging into the parabola formula 0.2619 for y we can calculate the t -coordinate :
t = 21.0 * 0.26 * 0.26 - 11.0 * 0.26 - 2.0
or t = -3.440
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : t = 21y2-11y-2
Axis of Symmetry (dashed) {y}={ 0.26}
Vertex at {y,t} = { 0.26,-3.44}
y -Intercepts (Roots) :
Root 1 at {y,t} = {-0.14, 0.00}
Root 2 at {y,t} = { 0.67, 0.00}
Solve Quadratic Equation by Completing The Square
8.2 Solving 21y2-11y-2 = 0 by Completing The Square .
Divide both sides of the equation by 21 to have 1 as the coefficient of the first term :
y2-(11/21)y-(2/21) = 0
Add 2/21 to both side of the equation :
y2-(11/21)y = 2/21
Now the clever bit: Take the coefficient of y , which is 11/21 , divide by two, giving 11/42 , and finally square it giving 121/1764
Add 121/1764 to both sides of the equation :
On the right hand side we have :
2/21 + 121/1764 The common denominator of the two fractions is 1764 Adding (168/1764)+(121/1764) gives 289/1764
So adding to both sides we finally get :
y2-(11/21)y+(121/1764) = 289/1764
Adding 121/1764 has completed the left hand side into a perfect square :
y2-(11/21)y+(121/1764) =
(y-(11/42)) • (y-(11/42)) =
(y-(11/42))2
Things which are equal to the same thing are also equal to one another. Since
y2-(11/21)y+(121/1764) = 289/1764 and
y2-(11/21)y+(121/1764) = (y-(11/42))2
then, according to the law of transitivity,
(y-(11/42))2 = 289/1764
We'll refer to this Equation as Eq. #8.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(y-(11/42))2 is
(y-(11/42))2/2 =
(y-(11/42))1 =
y-(11/42)
Now, applying the Square Root Principle to Eq. #8.2.1 we get:
y-(11/42) = √ 289/1764
Add 11/42 to both sides to obtain:
y = 11/42 + √ 289/1764
Since a square root has two values, one positive and the other negative
y2 - (11/21)y - (2/21) = 0
has two solutions:
y = 11/42 + √ 289/1764
or
y = 11/42 - √ 289/1764
Note that √ 289/1764 can be written as
√ 289 / √ 1764 which is 17 / 42
Solve Quadratic Equation using the Quadratic Formula
8.3 Solving 21y2-11y-2 = 0 by the Quadratic Formula .
According to the Quadratic Formula, y , the solution for Ay2+By+C = 0 , where A, B and C are numbers, often called coefficients, is given by :
- B ± √ B2-4AC
y = ————————
2A
In our case, A = 21
B = -11
C = -2
Accordingly, B2 - 4AC =
121 - (-168) =
289
Applying the quadratic formula :
11 ± √ 289
y = ——————
42
Can √ 289 be simplified ?
Yes! The prime factorization of 289 is
17•17
To be able to remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. second root).
√ 289 = √ 17•17 =
± 17 • √ 1 =
± 17
So now we are looking at:
y = ( 11 ± 17) / 42
Two real solutions:
y =(11+√289)/42=(11+17)/42= 0.667
or:
y =(11-√289)/42=(11-17)/42= -0.143
Two solutions were found :
- y = -1/7 = -0.143
- y = 2/3 = 0.667
How did we do?
Please leave us feedback.