Problem 20 - Prealgebra

Question

Solve each equation. Check your solution. $$12=5 k-8$$

Step-by-Step Solution

Answer

The solution is \( k = 4 \).

Step 1: Isolate the variable term

To solve the equation \(12 = 5k - 8\), we want to isolate the term with the variable \(k\). Start by adding 8 to both sides of the equation. \[ 12 + 8 = 5k - 8 + 8 \] This simplifies to: \[ 20 = 5k \]

Step 2: Solve for the variable

Now, divide both sides of the equation by 5 to solve for \(k\). This will help us find the value of \(k\). \[ \frac{20}{5} = \frac{5k}{5} \] This simplifies to: \[ k = 4 \]

Step 3: Check the solution

To ensure that \(k = 4\) is the correct solution, substitute \(4\) back into the original equation \(12 = 5k - 8\) and check if both sides are equal. \[ 12 = 5(4) - 8 \] Calculate the right side: \[ 12 = 20 - 8 \] \[ 12 = 12 \] Since both sides of the equation are equal, our solution is correct.

Key Concepts

Isolating Variables in Equations

When solving equations, the goal is often to find the value of the unknown variable. This usually involves isolating the variable on one side of the equation. Let's explore how we go about isolating variables.

• The first step is to remove any constants or coefficients attached to the variable. These are usually numbers that are added, subtracted, multiplied, or divided with the variable.
• In the given exercise, we start with the equation \(12 = 5k - 8\). Here, to isolate \(k\), we need to remove the 8 from the right-hand side.
• To achieve this, we add 8 to both sides of the equation, resulting in \(20 = 5k\). Now the terms without the variable are eliminated from the variable side.
• Next, the coefficient of \(k\) is 5, so we divide both sides by 5, further isolating \(k\) as \(k = 4\).

Isolating the variable may require different steps based on the equation structure, but the key principle is to perform inverse operations to simplify the equation.

Checking Solutions for Accuracy

After finding a solution for a variable, it is crucial to validate this solution by substituting it back into the original equation. This step is called "checking solutions" and ensures that the solution holds true.

• For our example, after determining that \(k = 4\), we substitute 4 back into the original equation \(12 = 5k - 8\).
• This requires computing the expression on the right-hand side: \(5(4) - 8\), which simplifies to \(20 - 8 = 12\).
• Once both sides of the equation are equal, it confirms that the value found for \(k\) is correct.

Checking solutions helps to avoid mistakes and ensures confidence that the problem has been solved accurately. Never skip this step to ensure understanding and correctness in solving equations.

Understanding Linear Equations

Linear equations are foundational in algebra and appear frequently in many mathematical applications. They are equations where the highest power of the variable is 1, giving them a distinct form and consistency.

• A standard linear equation is written in the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable.
• The exercise with \(12 = 5k - 8\) is a linear equation since \(k\) is raised to the first power only.
• These equations often graph as straight lines and have one solution, which is the point where the line crosses the x-axis.
• Understanding linear equations involves recognizing their formation and applying consistent steps to isolate variables and solve the equation.

Mastering linear equations provides a strong base for more advanced algebraic concepts, making them essential knowledge for any math student.