Problem 119 - Intermediate Algebra

Question

Find the value of \(k\) that makes 4 a solution of the following linear equation in \(x .\) $$ k+3 x-6=3 k x-k+16 $$

Step-by-Step Solution

Answer

The value of \( k \) is \(-1\).

Step 1: Understanding the Problem

We need to find the value of \( k \) that makes \( x = 4 \) a solution to the equation \( k + 3x - 6 = 3kx - k + 16 \). This means we will substitute \( x = 4 \) into the equation and solve for \( k \).

Step 2: Substitute \( x = 4 \)

Substitute \( x = 4 \) into the equation: \( k + 3(4) - 6 = 3k(4) - k + 16 \).

Step 3: Simplify Both Sides

Simplify both expressions: \( k + 12 - 6 = 12k - k + 16 \). This simplifies to \( k + 6 = 11k + 16 \).

Step 4: Rearrange the Equation

Subtract \( 11k \) from both sides to move all terms involving \( k \) to one side: \( k - 11k = 16 - 6 \).

Step 5: Combine Like Terms

Simplify both sides by combining like terms: \( -10k = 10 \).

Step 6: Solve for \( k \)

Divide both sides by \(-10\): \( k = \frac{10}{-10} = -1 \). Thus, the value of \( k \) is \(-1\).

Key Concepts

Solution of Linear Equations

Solving linear equations means finding the value of variables that make the equation true. In our exercise, we were tasked with determining the value of \( k \) such that \( x = 4 \) becomes a solution to the given linear equation. Linear equations typically involve expressions where variables do not have exponents. They are easy to manage due to their straightforward form. Here, the equation is: \[ k + 3x - 6 = 3kx - k + 16 \] To solve, we substitute the given value of \( x = 4 \) to focus on finding \( k \). By transforming the equation appropriately, we apply a series of mathematical operations to isolate and solve for \( k \). Understanding every step, while making changes to individual terms on each side, is crucial in mastering linear equations.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations and expressions to isolate a particular variable or term. This concept is at the heart of solving linear equations effectively. Let's break it down using our example. Initially, we have the modified equation: \[ k + 12 - 6 = 12k - k + 16 \] We simplify by combining like terms such that: - Left side: \( k + 12 - 6 = k + 6 \) - Right side: \( 12k - k + 16 = 11k + 16 \) Thus, the equation becomes: \[ k + 6 = 11k + 16 \] By algebraic manipulation, we move similar terms together, eventually isolating \( k \) by subtracting \( 11k \) from both sides of the equation. This helps in clearly seeing and executing the operations required to solve for \( k \).

Substitution Method

The substitution method is a technique used to solve equations where you replace variables with specific values. This is an essential step when verifying solutions or simplifying complex equations. In our exercise, we substituted \( x = 4 \) into the original equation. Substituting involves the following: - Replace every instance of \( x \) with \( 4 \): \[ k + 3(4) - 6 = 3k(4) - k + 16 \] - Simplify the numbers: - Left side becomes \( k + 12 - 6 = k + 6 \) - Right side simplifies to \( 12k - k + 16 = 11k + 16 \) These simplifications lead us to the straightforward equation \( k + 6 = 11k + 16 \). Substitution is effective because it reduces equations to a simpler form, making it easier to identify solutions, like the value of \( k = -1 \) in this case.