Problem 50 - Algebra A Combined Function

Question

Solve. $$ 15-(6-7 k)=2+6 k $$

Step-by-Step Solution

Answer

The solution is \(k = -7\).

Step 1: Simplify Inside the Parentheses

Start by simplifying the expression inside the parentheses on the left-hand side. \[15 - (6 - 7k) = 15 - 6 + 7k = 9 + 7k\]

Step 2: Rewrite the Equation

Now replace the simplified expression in the equation. The equation now looks like:\[9 + 7k = 2 + 6k\]

Step 3: Isolate the Variable Term

Subtract \(6k\) from both sides to bring the \(k\) terms together:\[9 + 7k - 6k = 2 + 6k - 6k\]This simplifies to:\[9 + k = 2\]

Step 4: Solve for k

Subtract 9 from both sides to isolate \(k\):\[k = 2 - 9\]This simplifies to:\[k = -7\]

Key Concepts

Solving Equations

When tackling a problem involving a linear equation like \(15-(6-7k)=2+6k\), the primary goal is to find the value of the variable that makes the equation true. In this case, the unknown variable is \(k\). Generally, solving linear equations involves a few systematic steps to help you arrive at the solution. First, simplify both sides of the equation, which may include expanding expressions, combining like terms, and getting rid of parentheses. Then, focus on organizing the equation so that all variable terms are on one side and constants on the other. The ultimate aim is to isolate the variable, making it easier to identify its value. By following these steps, you ensure that the equation is solved correctly and efficiently. Remember to always check your solution by substituting it back into the original equation to verify that it satisfies both sides.

Algebraic Manipulation

Algebraic manipulation is like solving a puzzle where pieces need to fit perfectly to uncover a message. In our equation, \(15-(6-7k)=2+6k\), we start our manipulation by addressing the parentheses.
Removing the parentheses means we take care of the signs and apply them correctly, leading to \(15 - 6 + 7k\). This step involves distributing negative signs across the terms inside the parentheses. Next, accurately combining constants or like terms helps simplify the equation further. After simplification, the equation turns into \(9 + 7k = 2 + 6k\), making it neater and easier to tackle.

• Distribute and simplify the expressions in all parentheses.
• Combine like terms, keeping constants and variables separate.
• Ensure that each step logically follows the previous one, maintaining the equation’s balance.

These manipulative techniques are crucial for converting complex equations into simpler forms, easing the journey toward finding the solution.

Variable Isolation

Variable isolation is crucial in solving equations. By isolating the variable, you find its value easily. In our case, simplify the equation further by moving all \(k\) terms to one side and constant terms to the other. This process is trickily named but can be quite straightforward:
Subtract \(6k\) from both sides to gather \(k\)-terms together, simplifying the equation to \(9 + k = 2\). This can be seen as neutralizing terms you don’t want with the variable. Next is the adjustment of constant terms. By subtracting 9 from both sides, you shift all constants to the other side, isolating \(k\):

• Ensure \(k\) terms stand alone on one side.
• Move constants adequately to solve for the variable \(k\).
• Use inverse operations to maintain equality.

As a result, you find \(k = -7\), which is the variable isolated and solved. Emphasizing such techniques fosters a clearer understanding and decreases errors in solving equations systematically.