Problem 10 - Fundamentals of Mathematics

Question

$$12 k-4=9 k-6+2 k$$

Step-by-Step Solution

Answer

k = -2

Step 1: Combine Like Terms

First, identify and combine the like terms on the right side of the equation: \(9k + 2k\). This will simplify the equation. \[12k - 4 = 11k - 6\]

Step 2: Isolate the Variable

Rearrange the equation to bring all terms containing \(k\) to one side. Subtract \(11k\) from both sides to isolate \(k\) on one side. \[12k - 11k - 4 = -6\] Which simplifies to \[k - 4 = -6\]

Step 3: Solve for the Variable

Now, solve for \(k\) by adding 4 to both sides of the equation. \[k - 4 + 4 = -6 + 4\] This simplifies to \[k = -2\]

Key Concepts

Combining Like Terms

When solving linear equations, combining like terms is a crucial first step. Like terms are terms in the equation that have the same variable raised to the same power. In the given example, both terms on the right side of the equation, \(9k\) and \(2k\), are like terms. Both have the variable \(k\) raised to the first power. To combine them, simply add their coefficients together. The coefficients here are 9 and 2, so adding them together gives \(11k\). This simplification results in an easier-to-manage equation:

• Original: \(12k - 4 = 9k - 6 + 2k\)
• Simplified: \(12k - 4 = 11k - 6\)

By simplifying expressions through combining like terms, we make the equation less cluttered and more straightforward to solve in subsequent steps.

Isolating the Variable

Once you have simplified the equation by combining like terms, the next step is to isolate the variable. Isolating the variable means rearranging the equation so that the variable \(k\) is alone on one side of the equation. This process involves moving all terms involving \(k\) to one side and constant terms to the other.In the example, you start by subtracting \(11k\) from both sides to get:

• \(12k - 11k - 4 = -6\)
• which simplifies to \(k - 4 = -6\)

By rearranging the terms, we further simplify the equation and isolate \(k\) on one side. Now, the equation can be solved much more easily because there's only one instance of \(k\). This step is crucial in making the variable the subject of the equation, preparing us to solve for its value.

Algebraic Manipulation

Finally, algebraic manipulation is used to solve for the variable once it has been adequately isolated. In our equation, we have arrived at \(k - 4 = -6\). The aim is to solve for \(k\) by performing operations that will eliminate any constants from the side with the variable.Here, you would add 4 to both sides to cancel out the \(-4\) next to \(k\). This gives us:

• \(k - 4 + 4 = -6 + 4\)
• which simplifies to \(k = -2\)

By performing these operations, we further manipulate the equation to isolate the variable completely, resulting in its exact value. These basic algebraic manipulations form the foundation of solving more complex equations and are essential skills in algebra.