Problem 66 - College Algebra Essentials

Question

Solve equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. \(5 x+7=2 x+7\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0\), and this equation is a conditional equation.

Step 1: Simplify the equation

Given the expression, subtract \(2x\) from both sides to get \(5x - 2x = 7 - 7\), which simplifies to \(3x = 0\).

Step 2: Solve for the variable

Divide both sides of the equation by 3 to isolate \(x\), getting \(x = 0\).

Step 3: Determine the type of the expression

After substituting \(x = 0\) back into the equation, both sides become equal creating a valid statement, thus it is a conditional equation as it is true for a specific value of \(x\), i.e., \(x = 0\).

Key Concepts

Conditional Equations

A conditional equation is one that is true only for specific values of the variable. This means the equation will not hold true for every possible value you might plug in for the variable. It's like having a key that fits only one lock — it will work perfectly for that specific lock but nowhere else.
For example, in the exercise provided, the equation was given as \(5x + 7 = 2x + 7\). By simplifying it, you discovered that \(x = 0\). When \(x\) is zero, both sides of the equation balance out perfectly, making it true. If you try any other number for \(x\), the equation would not be balanced anymore.
This indicates that \(5x + 7 = 2x + 7\) is a conditional equation because it holds true under the condition that \(x = 0\) only. So, whenever you solve an equation, if you find it is only valid for certain values, it's conditional.

Identity Equations

Identity equations are like magical equations that are always true, no matter what value you give the variable. When faced with an identity equation, substituting any number for the variable always results in a true statement. This is because both sides of the equation are equivalent expressions.
Let's consider a simple example. Take the equation \(3x + 5 = 3x + 5\). No matter what number you replace \(x\) with, both sides will always be equal. This doesn't rely on a specific number or condition but shows a universal truth for the structure of the equation itself.
The beauty of identity equations is in their simplicity—their accuracy doesn't waver. When solving equations, recognizing an identity equation means knowing you can plug any number into it and still keep it balanced. It's like discovering a universal truth in algebra.

Inconsistent Equations

Inconsistent equations present a unique scenario where they remain stubbornly false, no matter what value you assign to the variable. These equations create a situation where both sides can never be equal, similar to trying to fit a square peg in a round hole.
Imagine an equation like this: \(x + 4 = x + 7\). When you simplify it, you find yourself with a setup that demands an impossible equality, such as \(4 = 7\). Obviously, this can never be true no matter what number replaces \(x\).
Encountering an inconsistent equation in your algebraic adventures signals a dead-end where no solutions exist. It's an alert indicating that your journey to balance the equation is in vain, because these sides of the equation can just never meet in truth.