Problem 6 - Algebra 1: Concepts and Skills

Question

Solve the equation for the indicated variable. $$ 2 j+5=k ; j $$

Step-by-Step Solution

Answer

The solution of the equation for \( j \) is \( j = (k - 5) / 2 \).

Step 1: Understand the Goal

The goal is to solve for the variable \( j \) in the equation. That means getting \( j \) alone on one side of the equation.

Step 2: Isolate Terms With \( j \)

The variable \( j \) is currently in the term \( 2j \) and there is a +5 term on the same side of the equation. We have to move +5 to the other side of the equation to begin isolating \( j \). We do this by subtracting 5 from both sides of the equation. This gives the new equation \( 2j = k - 5 \).

Step 3: Solve for \( j \)

Now, \( j \) is in the term \( 2j \). To isolate \( j \), we need to get rid of this 2. Since it is being multiplied by \( j \), we use division to remove it. Dividing both sides of the equation by 2 gives: \( j = (k - 5) / 2 \).

Key Concepts

Isolating a Variable

When solving equations, one common task is to isolate the variable you're interested in. In our exercise, that variable is \( j \). Isolating a variable means that we manipulate the equation to get the variable alone on one side. This is crucial for finding its value.
To isolate \( j \) in the equation \( 2j + 5 = k \), we need to remove anything affecting it. The first step is to identify all terms attached to \( j \). Here, 5 is added to the term containing \( j \). By subtracting 5 from both sides, we ensure the equation remains balanced.

• Original Equation: \( 2j + 5 = k \)
• Subtract 5 from each side: \( 2j = k - 5 \)

Now \( j \) is more isolated since 5, which was on the same side, is gone.
Remember, whatever operation you perform on one side of the equation, you must also do it on the other side to maintain equality.

Solve for a Variable

After isolating the variable as much as possible, the next step is actually solving for it. This involves further operations to completely isolate the variable in terms of the other numbers or variables.
In the given problem, once we reached \( 2j = k - 5 \), \( j \) still wasn't completely isolated because it is multiplied by 2. To solve for \( j \), division is the right step.
You must divide every term in the equation by 2 to get:

• Equation before division: \( 2j = k - 5 \)
• Divide each side by 2: \( j = \frac{k - 5}{2} \)

By dividing, we achieve \( j \) alone on one side of the equation. Now, \( j \) is expressed in terms of \( k \), and it is completely isolated and solved.

Linear Equations

Linear equations are equations of the first degree, which means they graph as a straight line. In our exercise, the equation \( 2j + 5 = k \) is a linear equation because the variable \( j \) is to the power of 1.
These equations typically come in the form \( ax + b = c \), where \( a \), \( b \), and \( c \) can be any numbers. Here \( a = 2 \), \( b = 5 \), and \( c = k \).
Working with linear equations requires use of simple operations (addition, subtraction, multiplication, and division). These equations are fundamental because:

• They can represent many real-world problems where relationships between quantities are predictable and constant.
• Solving them helps build a foundation for understanding more complex algebraic expressions.

Learning to manipulate linear equations is a vital skill in algebra. This skill enables you to express relationships in a clear algebraic syntax and solve them efficiently.