When we solve a simple equation in algebra, we may have the following two situations.
It is possible that the equation may be true for all values of the variable. That is, the equation will have infinitely many solutions. Such an equation is called an identity.
It is also possible that the equation may have no solution. That is, there is no value of the variable that will result in a true equation.
When we solve a simple equation with infinitely many solutions or no solution, in the result we get at the last step, the variable will vanish. That is, there will be no variable.
If the result at the last step is true, then the equation has infinitely many solutions.
If the result at the last step is false, then the equation has no solution.
We can understand this stuff more clearly in the following examples.
Example 1 :
Solve the following equation :
2(1 - x) + 5x = 3(x + 1)
Solution :
2(1 - x) + 5x = 3(x + 1)
Simplify both sides.
2 - 2x + 5x = 3x + 3
2 + 3x = 3x + 3
Subtract 3x from each side.
2 = 3
The above result is false. Because 2 is not equal to 3. Because the result we get at the last step is false, the given equation has no solution.
Example 2 :
Solve the following equation :
(1/2)(8y - 6) = 5y - (y + 3)
Solution :
(1/2)(8y - 6) = 5y - (y + 3)
Simplify both sides.
4y - 3 = 5y - y - 3
4y - 3 = 4y - 3
Subtract 4y from each side.
-3 = -3
The above result is true. Because the result we get at the last step is true, the given equation has infinitely has many solutions.
Example 3 :
Solve the following equation :
(1/3)(9 - 6x) = 5 - 2x
Solution :
(1/3)(9 - 6x) = 5 - 2x
Simplify.
3 - 2x = 5 - 2x
Add 2x to each side.
3 = 5
The above result is true. Because the result we get at the last step is true, the given equation has infinitely has many solutions.
Example 4 :
(1/3)(15 - 6x) = 5 - ax
If the linear equation above is an identity, what is the value of a?
Solution :
(1/3)(15 - 6x) = 5 - ax
Simplify.
5 - 2x = 5 - ax
Because the given equation is an identity, the coefficients of like terms on both sides must be equal.
That is, coefficients of 'x' terms on the left side and right side must be equal.
So, equate the coefficients of 'x'.
- 2 = - a
Multiply each side by (-1).
a = 2
Example 5 :
In no value of x is the solution to the following equation, what is the value of a?
2ax - 15 = 3(x - 1) + 5(x + 2)
Solution :
2ax - 15 = 3(x - 1) + 5(x + 2)
Simplify.
2ax - 15 = 3x - 3 + 5x + 10
2ax - 15 = 8x + 7
From the given information, it is clear that the equation has no solution.
In the simplified equation above, if 2a = 8, then
8x - 15 = 8x + 7
Subtract 8x from both sides.
-15 = 7 (False)
In the given equation, if 2a = 8, then the equation has no solution.
2a = 8
Divide both sides by 2.
a = 4
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