Problem 92 - Intermediate Algebra

Question

Solve each equation. $$ 4 j+12.54=18.12 $$

Step-by-Step Solution

Answer

The solution is \( j = 1.395 \).

Step 1: Isolate the Term with the Variable

The given equation is \( 4j + 12.54 = 18.12 \). To begin solving for \( j \), we need to isolate the \( 4j \) term. To do this, subtract \( 12.54 \) from both sides of the equation:\[4j + 12.54 - 12.54 = 18.12 - 12.54\]This simplifies to:\[4j = 5.58\]

Step 2: Solve for the Variable

Now that we have \( 4j = 5.58 \), we need to solve for \( j \). To do this, divide both sides of the equation by 4:\[j = \frac{5.58}{4}\]Calculating the division gives:\[j = 1.395\]

Step 3: Verify the Solution

To ensure the solution is correct, substitute \( j = 1.395 \) back into the original equation to check:\[4(1.395) + 12.54 = 18.12\]Calculate:\[5.58 + 12.54 = 18.12\]Since both sides of the equation are equal, \( j = 1.395 \) is indeed the correct solution.

Key Concepts

Solving Equations

Solving linear equations involves finding the value of the variable that makes the equation true. In our example, the equation is given as \( 4j + 12.54 = 18.12 \). The key goal is to determine the value of \( j \) that satisfies the equation.

To begin, focus on isolating the variable term, which will help you identify the lone \( j \). Equations like these are often solved through a series of logical arithmetic steps. You will guide these actions by taking operations on both sides of the equation to maintain equality. Each step should bring you closer to writing the equation in a simpler, more recognizable form (e.g., \( j = \text{ something} \)). This process helps to break down complex real-world problems into manageable steps for straightforward computation.

Isolation of Terms

Isolation of terms is the fundamental step to simplifying and solving equations. In the equation \( 4j + 12.54 = 18.12 \), our primary task is to get the term with the variable, \( 4j \), on its own. This means we need to rid the left side of any other numbers not related to \( j \).

To do this, subtract \( 12.54 \) from both sides:

• Left Side: \( 4j + 12.54 - 12.54 = 4j \)
• Right Side: \( 18.12 - 12.54 = 5.58 \)

After subtraction, the equation reduces to \( 4j = 5.58 \). Here, we have successfully isolated the term with the variable, making it possible to move toward solving for \( j \).

Dividing both sides by 4, results in \( j = \frac{5.58}{4} \), which simplifies to \( j = 1.395 \).

Remember, isolation means getting our variable term alone, clearing any constant terms, so we can clearly and cleanly solve for the variable.

Verifying Solutions

After solving an equation, verification is crucial to ensure the solution is correct. Verification involves substituting the solution back into the original equation to see if it holds true.

For our example, we've found that \( j = 1.395 \). We substitute this back into the original equation \( 4j + 12.54 = 18.12 \):

• Replace \( j \) with 1.395: \( 4(1.395) + 12.54 \)
• Calculate \( 4 \times 1.395 = 5.58 \)
• Combine: \( 5.58 + 12.54 = 18.12 \)

Both sides of the equation are equal, confirming our solution \( j = 1.395 \) is correct.

This step is essential for building confidence in your solution. You're not just finding the answer, you're affirming that it fits perfectly into the original equation, just like a puzzle piece. Always make sure to check your work!