Problem 13 - Algebra A Combined Function

Question

Solve each equation. Check each solution. See Examples 1 through \(6 .\) \(\frac{k}{-7}=0\)

Step-by-Step Solution

Answer

The solution is \( k = 0 \).

Step 1: Identify Equation Type

The given equation is \( \frac{k}{-7} = 0 \). This is a simple linear equation, where \( k \) is divided by \( -7 \) and equals zero.

Step 2: Isolate the Variable

To solve \( \frac{k}{-7} = 0 \), multiply both sides of the equation by \( -7 \) to isolate \( k \).\[ -7 \cdot \frac{k}{-7} = 0 \cdot -7 \]

Step 3: Simplify

Upon simplifying, the left side reduces to \( k \) because multiplying \( -7 \) by \( \frac{k}{-7} \) cancels out the \( -7 \), and the right side remains \( 0 \).\[ k = 0 \]

Step 4: Verify the Solution

Substitute \( k = 0 \) back into the original equation to check.\[ \frac{0}{-7} = 0 \]Both sides equal zero, so the solution \( k = 0 \) is correct.

Key Concepts

Variable Isolation

In solving linear equations, one of the crucial steps is isolating the variable. This process involves arranging the equation so that the variable stands alone on one side, usually the left, making its value explicit. For instance, take the equation \( \frac{k}{-7} = 0 \). Here, the variable \( k \) is divided by \(-7\). To isolate \( k \), we must eliminate \(-7\) by performing the inverse operation, which is multiplication.

• Multiply both sides of the equation by \(-7\), targeting the denominator linked to the variable.
• This step ideally cancels the division when the same number divides and multiplies the variable.

After multiplying, the equation transforms to \( k = 0 \), successfully leaving \( k \) isolated, providing a clear view of its value.

Solution Verification

Verifying the solution is an essential step to ensure accuracy and correctness. Once you have isolated the variable and found its value, substitute back into the original equation to double-check.For our equation \( \frac{k}{-7} = 0 \), with the solution \( k = 0 \), plug it back:

• Substitute \( k = 0 \) into \( \frac{k}{-7} \).
• Calculate \( \frac{0}{-7} = 0 \).

Both sides equal zero, confirming that our solution is indeed correct. This verification step detects errors and reinforces understanding by ensuring the solution satisfies the original equation.

Equation Simplification

Simplification in solving equations involves reducing expressions to their simplest form, making it easier to identify solutions. In our context, simplification occurred when we multiplied \(-7\) by \( \frac{k}{-7} \).

• The multiplication cancels the division, simplifying the expression to \( k \).
• This leads directly to the variable holding its isolated, simplified form.

Further simplification isn't needed here because \( k \) already represents the simplest form, indicating the solution \( k = 0 \). Any unnecessary complexity has been removed, offering a clear and precise outcome.