Solving Linear Equations
The two examples below show that it is not true that the solution to every
linear equation is a unique real number.
Example 1
Solve: 4(x - 2) = 10x
+ 5 - 6x
| Solution |
4(x - 2) |
= 10x
+ 5 - 6x |
| Step 1 Remove parentheses.
Distribute the 4.
|
4x - 8 |
= 10x + 5 - 6x |
| Step 2 On each side of the equation, combine like
terms.
Combine 10x and -6x.
|
4x - 8
|
= 4x + 5 |
| Step 3 Isolate the variable.
Subtract 4x from both sides.
Simplify. |
4x - 8 - 4x
-8 |
= 4x + 5 - 4x
= 5 |
The result, -8 = 5 is a not a
true statement. So, there is no value of x which will make the original equation
true.
Therefore, the equation 4(x - 2) =
10x
+ 5 - 6x has no solution.
Note:
The statement -8 = 5 is a false statement,
which is sometimes called a contradiction.
Example 2
Solve: 3(y - 2) + 2y
= -6 + 5y
| Solution |
3(y - 2)
+ 2y |
= -6 + 5y |
| Step 1 Remove parentheses.
Distribute the 3.
|
3y - 6 + 2y |
= -6 + 5y |
| Step 2 On each side of the equation,
combine like terms.
Combine 3y and 2y.
|
5y - 6 |
= -6 + 5y |
| Step 3 Isolate the variable.
Subtract 5y from both sides.
Simplify. |
5y - 6 - 5y
-6 |
= -6 + 5y - 5y
-6 |
The result, -6 = -6 is a true statement. Therefore, any value of y will
make the original equation true.
So, the solution of 3(y - 2) + 2y
= -6 + 5y is all real numbers.
Note:
Here is another way to solve
5y - 6 = -6 + 5y.
| Add 6.
Divide by 5. |
5y
y |
= 5y
= y |
The statement y = y is true for all real
numbers.
The statement - 6 = -6 is a true
statement, which is sometimes called an identity.
|
