Problem 22 - College Algebra with Modeling and Visualization

Question

Solve the equation and check your answer. $$ 2 k-3=k+3 $$

Step-by-Step Solution

Answer

The solution to the equation is \( k = 6 \).

Step 1: Simplify the Equation

Start with the given equation: \( 2k - 3 = k + 3 \). Our first task is to isolate term with variable on one side. Subtract \( k \) from both sides to move all terms with \( k \) to the left side. This gives us: \( 2k - k - 3 = 3 \), which simplifies to \( k - 3 = 3 \).

Step 2: Solve for Variable

Now, with \( k - 3 = 3 \) from the previous step, isolate \( k \) by adding 3 to both sides of the equation. This results in \( k = 3 + 3 \), which simplifies to \( k = 6 \).

Step 3: Verify the Solution

To ensure that \( k = 6 \) is correct, substitute \( 6 \) back into the original equation: \( 2k - 3 = k + 3 \). Replace \( k \) with \( 6 \) to get \( 2(6) - 3 = 6 + 3 \). Simplify both sides: \( 12 - 3 = 9 \) and \( 6 + 3 = 9 \). Both sides are equal, confirming that \( k = 6 \) is correct.

Key Concepts

Solving Equations

Solving equations is all about finding the value of the unknown, often represented by a variable like \( k \). It involves rearranging and simplifying the equation until you isolate the variable on one side.
This process can be broken down into a series of steps:

• First, identify the terms that contain the variable and those that do not.
• Use operations like addition, subtraction, multiplication, and division to move these terms around.
• The ultimate goal is to have the variable by itself on one side and a number on the other.

In our example, we started with the equation \( 2k - 3 = k + 3 \). By subtracting \( k \) from both sides, we simplified it to \( k - 3 = 3 \). This step is crucial as it makes the equation simpler and closer to revealing the value of \( k \).
Once you've isolated the variable, you can solve for it directly. In this example, adding 3 to both sides gave us \( k = 6 \). This straightforward process applies to many algebraic equations.

Linear Equations

Linear equations are a special type of equation where the variable takes on the form of a straight line when graphed.
They are called "linear" because they represent a line, and they have the general format of \( ax + b = c \).
Key characteristics of linear equations include:

• The variable is raised only to the first power.
• There are no products or roots of variables involved.
• They have constant ratios, meaning they distribute evenly across the line represented.

In our solved equation \( 2k - 3 = k + 3 \), after simplification, we have \( k = 6 \) which perfectly fits the structure of a linear equation.
Understanding these properties helps in identifying and solving linear equations faster, as it establishes a predictable pattern that can be applied across different problems.

Verification of Solutions

Verification of solutions in algebra ensures that the answer you obtained is indeed correct. It acts as a proof of correctness for the solution derived from the equations.
Verification can be done by substituting the obtained solution back into the original equation and checking if the equation holds true.
In our example, after solving for \( k = 6 \), we substitute this back into the original equation \( 2k - 3 = k + 3 \). When calculating,

• The left side evaluates to \( 12 - 3 = 9 \).
• The right side evaluates to \( 6 + 3 = 9 \).
• Both sides match, confirming the solution is correct.

This step ensures there are no calculation errors or logical fallacies in the process of solving equations.
It's a critical step that provides confidence in your solution and is an essential practice in all of algebra.