Problem 95 - Algebra and Trigonometry

Question

Solve and determine whether $$8(x-3)+4=8 x-21$$ is an identity, a conditional equation, or an inconsistent equation.

Step-by-Step Solution

Answer

The equation \(8(x-3)+4=8x-21\) is an inconsistent equation.

Step 1: Distribute the multiplication

Start by distributing the multiplication over subtraction in the left-hand side of the equation: \(8(x - 3) + 4 = 8x -21\). This results in \(8x - 24 + 4 = 8x - 21\)

Step 2: Simplify each side of the equation

Next, simplify each side of the equation by combining like terms, if any. This gives \(8x - 20 = 8x - 21\)

Step 3: Check if the equation is identity, inconsistent, or conditional

If you subtract \(8x\) from both sides, the equation becomes \(-20 = -21\). Since this statement is false, the original equation cannot be an identity (an equation that is always true regardless of the value of \(x\)), nor a conditional equation (an equation that is true for certain values of \(x\)). Therefore, the original equation is an inconsistent equation, which means it has no solution.

Key Concepts

Identity Equation

An identity equation is a special type of equation that is always true, no matter what value is substituted for the variable. When you solve an equation, if the variable disappears leaving a true statement, then you have an identity equation.
Consider a simple example of an identity equation:

• \(x + 3 = x + 3\)

If you subtract \(x\) from both sides, you're left with:

• \(3 = 3\)

Since this is always true, the equation is an identity. Identity equations can help reinforce concepts in mathematics because they always produce true statements, signaling that every number in the domain of the variable will satisfy the equation.
To identify an identity equation, simplify it until all variables are eliminated and a true statement remains.

Conditional Equation

A conditional equation is one that is true only for specific values of the variable. Unlike identity equations, conditional equations are not true for all values and have a specific solution.
For example, consider the equation:

• \(2x + 3 = 11\)

To find the solution, subtract 3 from both sides to get:

• \(2x = 8\)

Then divide by 2:

• \(x = 4\)

This means the equation is only true when \(x = 4\). That's the conditional solution. Conditional equations are prevalent in problem-solving scenarios where a specific answer is needed, such as in real-world applications involving measurements or finances.
If you end up with a variable equal to a number after simplifying, you have a conditional equation.

Inconsistent Equation

An inconsistent equation is an equation that has no solution. This occurs when, after simplifying, you end up with a false statement, such as when the two sides of the equation do not equal each other and there are no values for the variable that will make it true.
Taking the initial exercise example:

• \(8(x - 3) + 4 = 8x - 21\)

After simplifying, you end up with:

• \(-20 = -21\)

This clearly doesn't make sense, as \(-20\) never equals \(-21\). When this happens, we call it inconsistent. Inconsistent equations often occur in equations derived from incorrect assumptions or when two linear equations represent parallel lines that will never intersect.
Always check your work. If after simplification you are left with a contradiction, your equation is inconsistent, denoting there is no value of the variable that can satisfy the equation.