Classifying Equations
Classifying Equations
Consider the equation we solved at the start of the last section, \(7x+8=-13\). The solution we found was \(x=-3\). This means the equation \(7x+8=-13\) is true when we replace the variable, x, with the value \(-3\). We showed this when we checked the solution \(x=-3\) and evaluated \(7x+8=-13\) for \(x=-3\).
If we evaluate \(7x+8\) for a different value of x, the left side will not be \(-13\).
The equation \(7x+8=-13\) is true when we replace the variable, x, with the value \(-3\), but not true when we replace x with any other value. Whether or not the equation \(7x+8=-13\) is true depends on the value of the variable. Equations like this are called conditional equations.
All the equations we have solved so far are conditional equations.
Conditional equation
An equation that is true for one or more values of the variable and false for all other values of the variable is a conditional equation.
Now let’s consider the equation \(2y+6=2\left(y+3\right)\). Do you recognize that the left side and the right side are equivalent? Let’s see what happens when we solve for y.
| Distribute. | |
| Subtract \(2y\) to get the \(y\)’s to one side. | |
| Simplify—the \(y\)’s are gone! |
But \(6=6\) is true.
This means that the equation \(2y+6=2\left(y+3\right)\) is true for any value of y. We say the solution to the equation is all of the real numbers. An equation that is true for any value of the variable like this is called an identity.
Identity
An equation that is true for any value of the variable is called an identity.
The solution of an identity is all real numbers.
What happens when we solve the equation \(5z=5z-1\)?
| Subtract \(5z\) to get the constant alone on the right. | |
| Simplify—the \(z\)’s are gone! |
But \(0\ne \text{−}1\).
Solving the equation \(5z=5z-1\) led to the false statement \(0=-1\). The equation \(5z=5z-1\) will not be true for any value of z. It has no solution. An equation that has no solution, or that is false for all values of the variable, is called a contradiction.
Contradiction
An equation that is false for all values of the variable is called a contradiction.
A contradiction has no solution.
Example
Classify the equation as a conditional equation, an identity, or a contradiction. Then state the solution.
\(6\left(2n-1\right)+3=2n-8+5\left(2n+1\right)\)
Solution
| Distribute. | |
| Combine like terms. | |
| Subtract \(12n\) to get the \(n\)’s to one side. | |
| Simplify. | |
| This is a true statement. | The equation is an identity. The solution is all real numbers. |
Example
Classify as a conditional equation, an identity, or a contradiction. Then state the solution.
\(10+4\left(p-5\right)=0\)
Solution
| Distribute. | |
| Combine like terms. | |
| Add \(10\) to both sides. | |
| Simplify. | |
| Divide. | |
| Simplify. | |
| The equation is true when \(p=\frac{5}{2}\). | This is a conditional equation. The solution is \(p=\frac{5}{2}.\) |
Example
Classify the equation as a conditional equation, an identity, or a contradiction. Then state the solution.
\(5m+3\left(9+3m\right)=2\left(7m-11\right)\)
Solution
| Distribute. | |
| Combine like terms. | |
| Subtract \(14m\) from both sides. | |
| Simplify. | |
| But \(27\ne -22\). | The equation is a contradiction. It has no solution. |
| Type of equation | What happens when you solve it? | Solution |
|---|---|---|
| Conditional Equation | True for one or more values of the variables and false for all other values | One or more values |
| Identity | True for any value of the variable | All real numbers |
| Contradiction | False for all values of the variable | No solution |
This lesson is part of:
Solving Linear Equations II