Problem 44 - Elementary Algebra

Question

Solve. $$ -3 y+2=-13 $$

Step-by-Step Solution

Answer

The solution is \(y=5\).

Step 1: Isolate the Variable Term

Start by isolating the term with the variable on one side of the equation. You can do this by subtracting 2 from both sides of the equation:\(-3y + 2 - 2 = -13 - 2\)This simplifies to:\(-3y = -15\)

Step 2: Solve for the Variable

Now that \(-3y = -15\), you need to solve for \(y\) by dividing each side by \(-3\):\[y = \frac{-15}{-3}\]This simplifies to:\(y = 5\)

Key Concepts

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra. It involves finding the value of the variable that makes the equation true. Linear equations are usually in the form of \( ax + b = c \), where \( x \) is the variable we want to find.
To solve these equations, we perform operations that simplify and eventually isolate the variable. The operations must maintain balance in the equation. This means doing the same thing to both sides of the equation
Common operations include:

• Addition or subtraction to move terms from one side to the other
• Multiplication or division to simplify coefficients of the variable

By understanding and mastering these operations, you're able to solve simple linear equations and develop a foundation for more complex algebraic tasks.

Algebraic Manipulation

Algebraic manipulation involves restructuring an equation to make solving easier. It’s like reorganizing the equation into a form we can work with better.
Methods often used in algebraic manipulation include:

• Combining like terms: Group similar variable terms or constant terms together.
• Transposition: Move terms across the equality sign to isolate the variable. Change their sign while shifting from one side to the other.

Being comfortable with these techniques helps with keeping control over your work as you simplify an equation. By using algebraic manipulation effectively, you maintain balance while transforming equations into simpler forms, paving the way to find the value of the variable with less hassle.

Variable Isolation

Variable isolation is a key goal when solving equations. The idea is to get the variable alone on one side of the equation so you can read off its value plainly.
To isolate the variable, use the following steps:

• Remove constants from the variable side by adding or subtracting them.
• Divide or multiply to cancel out any coefficient in front of the variable.

For instance, in the original problem, we started by removing the constant (2) by subtracting it from both sides: \(-3y + 2 - 2 = -13 - 2\) simplifies to \(-3y = -15\).
Next, we divide both sides by the coefficient of the variable, \(-3\), resulting in \(y = \frac{-15}{-3}\). Therefore, \(y = 5\).
These systematic steps allow you to reach a conclusion while ensuring that your handling of the equation is methodical and accurate.