1 - When you write
3 + 7 = 7 + 3, you get a
numerical equality.
You can also check it quickly, by performing the calculations indicated on both sides of the = sign and making sure that the results found are the same.
2 - If you now write 3x + 7 = 7 + 3x, you get a
literal identity (note that by replacing (x) with any numerical value, you find a numerical equality).
Definition of identity:An identity is a literal equality that is verified for all numerical values that can be given to the letters it contains.
Example: Let's take the identity 3x + 7 = 7 + 3x and give (x) the following values :
x = 1 gives us 3 + 7 = 7 + 3
x = - 1 gives us - 3 + 7 = 7 - 3
x = 10 gives us 30 + 7 = 7 + 30....
Let's also quote : (a + b)2 = a2 + 2ab + b2 ....
3 - If you now write 2 (x + 6) = 4 x + 6, you may believe at first sight that it is false, and that would be false indeed, if you had wanted to write an identity. The above equality is checked only for a particular value of x. It's an equation. If you give the value 3 for x, the equation is solved and you have found a numerical equality. No other value of x solves this equation.
Definition:An equation is a literal equality that is verified only for certain values attributed to the letters it contains.
These values are the solutions or roots of the equation.
To solve an equation is to find all these roots (or solutions). In complex equations, several solutions can answer the problem and solve the equation. We will not study the so-called second or third degree equations, which would go beyond the scope of our lesson, which aims, as we have said, to make it easier for you to understand the different theories.
4. 2. - RESOLUTION OF EQUATIONS
To solve an equation, one must try to replace it with a simpler equivalent equation, whose roots are known.
Rule 1:When adding or subtracting the same algebraic expression to both members of an equation, we obtain an equation equivalent to the first one.
Let ------> 3 (x + 2) = 2x + 3
We can add (or subtract) to the two members, an algebraic number "A" any without changing the equation.
We can write : 3 (x + 2) + A = 2x + 3 + A
"A" can be a pure number, or contain the unknown x.
Remark:
A term can be transposed from one member of an equation into the other provided that the sign is changed.
Example :
3 + 2x = 3x + 4
Retrench 3 in each member :
3 + 2x - 3 = 3x + 4 - 3
Perform the possible operations in the first member ; he stays :
2x = 3x + 4 - 3
We observe that the term 3, which had the sign + in the first member, has the minus sign in the second, which confirms the stated rule.
So let's apply this rule for the 3x term :
2x - 3x = 4 - 3
- x = 1
or x = - 1
Remark: In the final result, always say "x positive". In the case where we obtain "x negative", it suffices to multiply the two terms of the equation by -1 to obtain "x positive".
Rule 2:
When one multiplies or divides the two members of an equation by the same non-zero algebraic expression, one obtains an equation equivalent to the first one.
A, B
and C
being algebraic expressions.
APPLICATIONS :
1 - We can "drive out the denominators" of an equation by multiplying one of its members by the denominator of the other and vice versa.
Example :
5x - 11 / 4 = 3x + 20 / 15
To "chase" the denominators, one multiplies by 15 the member of left and by 4 the member of right :
(5x - 11) . 15 = (3x + 20) . 4
Note : This reminds us of the property of the proportions "produces extremes = produces means".
The solution is :
75x - 165 = 12x + 80
63x = 245
x = 245 / 63
Another example :
Is : 5x - 11 / 4 = 3x + 20
Multiply by 4 the right limb. 4 "is chased" from the left limb and comes to multiply the right limb.
5x - 11 = 4 (3x + 20)
The resolution of the equation continues, then, as we have seen above :
5 x - 11 = 12x + 80
- 91 = 7x
from where : x = - 91 / 7 = - 13
2 - We can simplify an equation by dividing the two members by the same non-zero number.
With C, not zero. (Other property of the proportions).
Note:
Example de calcul :
We have the following equation :
(7x + 3) (x - 2) = (4x - 1) (x - 2)
We note the presence of (x - 2) in both members. We can therefore simplify them by x - 2 according to rule 2. But this expression (x - 2) vanishes for x = 2. The two members are then multiplied by 0 and the equation is verified. (We get 0 = 0).
The number 2 is a first root. We can now simplify by (x - 2). We have :
7x + 3 = 4x - 1
from where :
7x - 4x = - 3 - 1
3x = - 4
x = - 4 (4 / 3) which is the 2nd root
So we found in this equation the two roots :
x = 2 and x = - (4 / 3)
You see from this example, that you should be careful not to simplify without precaution, by an expression that can be canceled: you run the risk of removing a solution to the equation.
3 - The unknowns are passed to the left and the numerical values to the right.
2x2 - 2x2 + x - 4x + 5x = 2
2x = 2
x = 1
We now make sure that the found square root does not cancel a denominator :
We can confirm that 1 is the root of the equation.
Remark : The presence of 2x2indicates a second degree equation.
In this case, the term in x2
disappears by calculation, because it exists in both members. We return to a 1st degree equation that we know how to solve. If on the other hand after simplification, a term in x2 had remained present, we could not have solved the equation that did not study the equations of the second degree.
5. 2. -EQUATION WITH LITERAL COEFFICIENTS
Let's solve :
(ax - c) / b = (c - ax) / a
This equation only makes sense if the terms a, b and c are different from zero.
1 - Let's chase the denominators: :
(ax - c) a = (c - ax) b
a2x - ac = bc - abx
2 - Let's pass the terms in (x) in one member, the terms without (x) in the other.
a2x + abx = ac + bc
3 - Let's put (x) in common factor
x (a2 + ab) = c (a + b)
Let's now put "a" in common factor in the first member :
ax (a + b) = c (a + b)
4 - To obtain x alone, we must divide the first term by a (a + b). But to maintain equality, divide the two members by this value :
5 - Discussion :
If a + b is different from 0, we can simplify by a + b and we get :
x = c / a
If a + b is equal to 0, the equation is of the form :
ax . 0 = c . 0
He is indeterminate. Indeed, the two members are always null no matter what value is given to x.
These calculations may seem complicated. It is not so. These are just applications of basic rules that we have studied so far. On the other hand, it is very important to carefully observe an equation before developing the calculations.
5. 3. - RESOLUTIONS OF EQUATIONS
5. 3. 1.-BY FACTOR PRODUCTS
(3 - x) (x + 4) = 0
It is obvious that for a factor product to be zero, it is necessary and sufficient that one of the factors be zero.
Example :
107 x 109 x 0 = 0
Hence two possibilities and two solutions :
1)
3 - 4 = 0
that is to say
x = 3
2)
x + 4 = 0
that is to say
x = - 4
5. 3. 2. -BY COMMON FACTOR
Let the equation :
2x2 - 3x = 21x
2x2 - 3x - 21x = 0
2x2 - 24x = 0
Let x be a common factor :
x (2x - 24) = 0
For a product of factors to be zero, it is necessary and sufficient that one of the factors be zero.
Hence both solutions :
1)
x = 0
x = 0
2)
2x - 24 = 0
x = 12
5. 3. 3. - EQUATION CONTAINING FACTOR UNKNOWN IN BOTH MEMBERS
(x + 3) (2x - 2) = (x + 4) (x + 3)
We can see the term (x + 3) in both members. We encountered such a problem earlier.
If the term (x + 3) is zero, the equation is verified. This is the first solution.
x = - 3
If (x + 3) is not zero, we can simplify the two members by (x + 3) and we find :
First degree equations to an unknown
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