Problem 23 - Elementary Algebra

Question

Solve. $$ 23 y+12=58 y+3724 $$

Step-by-Step Solution

Answer

\(y \approx -106.06\)

Step 1: Move Variables to One Side

Begin by subtracting \(23y\) from both sides of the equation to move the variable terms to one side: \[23y + 12 - 23y = 58y + 3724 - 23y\] This simplifies to: \[12 = 35y + 3724\]

Step 2: Isolate the Variable Term

Subtract 3724 from both sides to isolate the \(y\) term: \[12 - 3724 = 35y + 3724 - 3724\] This simplifies to: \[-3712 = 35y\]

Step 3: Solve for \(y\)

Divide both sides by 35 to solve for \(y\): \[y = \frac{-3712}{35}\] So the value of \(y\) is approximately \(-106.06\).

Key Concepts

Linear Equations

A linear equation is a type of mathematical expression that involves variables raised to the power of one. These equations create a straight line when graphed on a coordinate plane. In the equation given for this exercise: \[ 23y + 12 = 58y + 3724 \]we see terms with the variable \(y\). The linear term here is \(23y\) and \(58y\), which tells us about the number of times \(y\) is being added. Linear equations are crucial because they model relationships where change is constant, providing a foundation for understanding more complex equations and scenarios.

Variable Isolation

Variable isolation is the process of rearranging an equation in order to make one variable the subject. This means getting the variable by itself on one side of the equation. In our example, we aim to solve for \(y\), meaning we want to get \(y\) alone: \[ 12 = 35y + 3724 \]The goal here is to perform operations that eliminate other terms, step by step:

• Adding or subtracting terms from both sides to remove constants.
• Dividing or multiplying both sides to cancel out coefficients with the variable.

To effectively isolate the variable, always remember the balance principle: what you do on one side, do on the other.

Step-by-Step Solution

Solving an equation using a step-by-step approach helps in maintaining clarity and reduces errors. Here's how to solve our given equation in steps:
1. Move the variables to one side by subtracting \(23y\) from both sides: \[ 12 = 35y + 3724 \] This organizes terms, making it easier to work with.
2. Isolate \(y\) by removing any constants: Subtract 3724 from both sides: \[ -3712 = 35y \]
3. Solve for \(y\) by dividing by the coefficient of \(y\), which is 35: \[ y = \frac{-3712}{35} \]This breakdown ensures each operation is clear, reinforcing understanding.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to rearrange and simplify equations. This includes the skills needed for moving terms across the equation, factoring, expanding expressions, and simplifying. In our exercise, we used algebraic manipulation by:

• Subtracting \(23y\) from both sides to consolidate variable terms.
• Subtracting 3724 to isolate the term with the variable \(y\).
• Lastly, dividing by 35 to find the exact value of \(y\).

These manipulations ensure we follow systematic approaches to reach a solution without altering the balance or integrity of the original equation.