Problem 14 - Prealgebra

Question

Solve equation. Check your solution. \(13 y-18=-5 y+36\)

Step-by-Step Solution

Answer

The solution to the equation is \(y = 3\).

Step 1: Move Terms Involving y to One Side

Start by adding \(5y\) to both sides of the equation to gather terms containing \(y\) on one side. This will give us:\[ 13y + 5y - 18 = 36 \] which simplifies to:\[ 18y - 18 = 36 \]

Step 2: Move Constant Terms to the Other Side

Add 18 to both sides of the equation to move constant terms to the right side. This results in:\[ 18y - 18 + 18 = 36 + 18 \] which simplifies to:\[ 18y = 54 \]

Step 3: Solve for y

Divide both sides of the equation by 18 to solve for \(y\):\[ y = \frac{54}{18} \]which simplifies to:\[ y = 3 \]

Step 4: Verify the Solution

Substitute \(y = 3\) back into the original equation to check:\(13(3) - 18 = -5(3) + 36\)Calculate both sides:\(39 - 18 = -15 + 36\)\(21 = 21\)The left and right sides are equal, confirming that the solution is correct.

Key Concepts

Solving Equations

Solving linear equations is a process of finding the value of the unknown variable that makes the equation true. Let's use the equation we have: \(13y - 18 = -5y + 36\). The goal is to isolate \(y\) on one side of the equation. This typically involves several key steps:

• Combining Like Terms: We begin by adding, subtracting, multiplying, or dividing terms to bring similar terms together. For example, by adding \(5y\) to both sides, we aim to gather all \(y\) terms on one side.
• Balancing the Equation: Whatever operation is performed on one side must also be done on the other side to keep the equation balanced. Thus, after obtaining \(18y - 18 = 36\), we maintain balance by adding \(18\) to both sides.
• Isolating the Variable: The final step is to solve for the variable by dividing both sides by the coefficient of \(y\), which is \(18\) in this example. This gives us \(y = 3\).

These steps ensure that the value obtained, when substituted back into the original equation, satisfies both sides.

Step-by-step Solution

Breaking down a solution into manageable steps can make solving equations much more approachable. Doing this helps to clarify how we handle each part of the equation. Here's how it unfolds for our problem:

• Step 1: Combine all terms involving \(y\) by adding \(5y\) to both sides. This gives us \(18y - 18 = 36\).
• Step 2: Move the constant term \(-18\) to the other side by adding \(18\) to both sides, resulting in \(18y = 54\).
• Step 3: Divide each side by \(18\) to isolate the variable \(y\), simplifying to \(y = 3\).

Taking it step-by-step emphasizes the logical progression and arithmetic behind solving linear equations. Each step should be understood thoroughly before moving to the next, as errors can propagate if the foundation is shaky.

Verifying Solutions

Verifying solutions is a crucial step in solving equations. It ensures that the solution is correct and that no mistakes were made. To verify, substitute the found value of the variable back into the original equation and check if both sides equal:

• Substitution: Replace \(y\) with \(3\) in the original equation to see if both sides match. That is, check \(13(3) - 18 = -5(3) + 36\).
• Calculate: Simplifying both sides yields \(21 = 21\), which verifies the solution is accurate.

By going through the verification step, we confirm that \(y = 3\) is indeed the solution. This validation technique is useful not only for catching potential errors but also for reinforcing understanding of the equation's structure.