Problem 43 - Algebra 1: Concepts and Skills

Question

Check to see if the given value of the variable is or is not a solution of the equation. \(8 k-2=30 ; k=4\)

Step-by-Step Solution

Answer

The given value \(k = 4\) is a solution to the equation \(8k - 2 = 30\).

Step 1: Understanding the equation

We have an equation \(8k - 2 = 30\), and we are given that \(k = 4\). The goal is to check if substituting \(k = 4\) in the equation yields a true statement.

Step 2: Substitute the value of k into the equation

Let's substitute \(k = 4\) into the equation. That gives us \(8(4) - 2 = 30\).

Step 3: Simplify the equation

Now, let's do the multiplication and subtraction on the left hand side of the equation: \(8(4) - 2 = 32 - 2 = 30\).

Step 4: Check if both sides of the equation are equal

We find that both sides of the equation are equal (30 = 30), so \(k = 4\) is indeed a solution to this equation.

Key Concepts

Substitution

Substitution in solving equations is like trying a key in a lock to see if it fits. We are given an equation, which in this case is \(8k - 2 = 30\), and a value for the variable, \(k = 4\). Our task is to "try" this value of \(k\) in the equation to determine if it works. This process involves replacing the variable, \(k\), with the given value in the entire equation. By doing this, we can transform the equation from:

• \(8k - 2 = 30\)
• to \(8(4) - 2 = 30\)

Remember, substitution is useful because it allows us to check whether or not the equation holds true when the variable is replaced with a specific value. By practicing this method, you'll enhance your confidence in checking for solutions.

Simplification

Once we substitute \(k = 4\) into the equation \(8(4) - 2 = 30\), the next step is to simplify the expression. Simplification means carrying out the arithmetic operations so that we can compare both sides of the equation easily.

Steps for Simplification:

• Perform multiplication: \(8 \times 4\) becomes \(32\).
• Next, complete the subtraction in the expression: \(32 - 2\) becomes \(30\).

Reaching a simpler form like \(30\) on the left side is crucial. Simplification helps in making the equation easy to interpret, and checking is straightforward when your arithmetic steps are correctly followed. Practice simplifying accurately to avoid errors and enhance your problem-solving efficiency.

Checking Solutions

After substitution and simplification, the most critical step is checking if the solution holds true for the entire equation. This is where we verify if both sides of the equation are equal after the substitution.

• For our equation \(8k - 2 = 30\), after substituting \(k = 4\) and simplifying to \(30 = 30\), observe that both sides match.
• This means the value \(k = 4\) is correct and solves the equation.

It is important because even a minor mistake in arithmetic can lead to incorrect solutions. Checking solutions validates our answer by confirming the equality of both sides post-substitution. This step strengthens understanding and assurance in the accuracy of the solving method.