Problem 31 - Elementary Algebra

Question

For the following problems, solve each conditional equation. If the equation is not conditional, identify it as an identity or a contradiction. $$ \frac{k}{15}+20=10 $$

Step-by-Step Solution

Answer

Answer: -150

Step 1: Write down the given equation

We have the equation: $$ \frac{k}{15} + 20 = 10 $$

Step 2: Isolate the term with the variable 'k'

To isolate the term with the variable 'k', we need to subtract 20 from both sides of the equation: $$ \frac{k}{15} + 20 - 20 = 10 - 20 $$ Which simplifies to: $$ \frac{k}{15} = -10 $$

Step 3: Solve for 'k'

Now that we have isolated the term with the variable 'k', we can solve for 'k' by multiplying both sides of the equation by 15: $$ 15 \cdot \frac{k}{15} = -10 \cdot 15 $$ The 15 on the left side of the equation cancels out with the denominator, leaving: $$ k = -150 $$

Step 4: Verify that the equation is conditional and state the solution

Since we were able to find a unique value for 'k' that makes the equation true, it is a conditional equation. The solution to the given equation is: $$ k = -150 $$

Key Concepts

Conditional Equations

Conditional equations are equations that are true only for specific values of the variable involved. They do not hold true for every value, just particular solutions, which makes them conditional. For example, in the exercise given, the equation \( \frac{k}{15} + 20 = 10 \) is only true when \( k = -150 \). If you substitute any other value for \( k \), the equation wouldn't be equal on both sides. This type of equation is quite common in algebra, where you're tasked with finding the precise value or values that satisfy the balance on either side of the equation.

• Identify by finding one unique solution.
• Check by substituting the solution back into the original equation to verify it holds true.

Conditional equations contrast with identities, which are always true no matter what value is substituted for the variable.

Identities

Identities refer to equations that are true for all possible values of the variable involved. These are expressions that balance perfectly no matter what number replaces the variable.
Unlike conditional equations, identities do not rely on finding a specific solution. They are universal truths, like \( a + b = b + a \), which illustrates the commutative property of addition.

• Always holds true, create no limit on variable values.
• Broadly applicable across various scenarios.

Identifying identities means recognizing patterns or rules within expressions that make them universally valid. In the context of algebra, learning to recognize identities is essential for simplifying expressions and solving complex problems.

Contradictions

Contradictions in algebra are equations that do not have any solutions. No matter what value you plug into the variable, the equation never balances. This means, mathematically, the left side and the right side can never be equal. A simple example may be \( x + 2 = x \).
Upon simplifying, you would end up with an untrue statement like \( 2 = 0 \), indicating there are no valid solutions.

• Leads to logical impossibilities.
• No solution exists that can satisfy the equation.

Contradictions can arise from miscalculations or errors in formulating algebraic expressions, and recognizing them helps in reframing approaches to problem-solving.

Algebraic Manipulation

Algebraic manipulation involves using algebraic rules and operations to rearrange or simplify equations or expressions in order to isolate variables and solve for unknowns. This process is crucial in handling all types of equations, whether they are conditional, identities, or contradictions.
For instance, in the provided exercise, algebraic manipulation was key to isolating \( k \):

• Subtracting \( 20 \) from both sides simplified the equation.
• Multiplying by \( 15 \) solved for the variable.

By mastering algebraic manipulation, one can efficiently find solutions, validate equations, or identify mathematical properties within expressions. It's an essential skill in mathematics that builds a foundation for advanced problem solving and critical thinking.