Problem 51 - Elementary Algebra

Question

Solve. $$ 3-2 y=-11 $$

Step-by-Step Solution

Answer

The solution is \( y = 7 \).

Step 1: Isolate the variable term

To solve for the variable \( y \), we need to isolate terms involving \( y \) on one side of the equation. Start by subtracting 3 from both sides of the equation to remove the constant from the left side: \[-2y = -11 - 3\]Which simplifies to:\[-2y = -14\]

Step 2: Solve for the variable

To solve for \( y \), divide both sides of the equation by -2 to isolate \( y \):\[y = \frac{-14}{-2}\]Calculating the division gives:\[y = 7\]

Key Concepts

Isolating Variables

When solving linear equations, one of the primary goals is to isolate the variable. This process involves manipulating the equation so that the unknown variable (in this case, \( y \)) stands alone on one side of the equation. By doing so, we can directly solve for its value. The first step generally involves removing constants or coefficients that are on the same side as the variable.
For example:

• In the equation \( 3 - 2y = -11 \), our goal is to get \( y \) by itself.
• We first eliminate the constant by subtracting 3 from both sides, resulting in \( -2y = -14 \).

As this isolation is achieved, it prepares the equation for further simplification using other operations like division or multiplication.

Integer Operations

Integer operations are mathematical procedures involving whole numbers, which include negative numbers, positive numbers, and zero. These operations are crucial when solving equations, especially when they involve isolating variables. In the context of this problem, integer operations help manage the terms in the equation, particularly when simplifying both sides.

• For instance, subtracting 3 from -11 involves understanding how to handle negative integers: \( -11 - 3 = -14 \).
• Furthermore, when \( -2y = -14 \), dividing both sides by -2 requires careful handling of negative numbers using division rules.

Integer operations ensure every step respects the rules of arithmetic, maintaining accuracy and consistency in solving equations.

Algebraic Expressions

Algebraic expressions form the building blocks of equations and involve terms that include numbers, variables, and operational symbols (+, -, *, /). When working on an equation, recognizing and simplifying these expressions are integral parts of reaching the solution.

• Initially, the equation \( 3 - 2y = -11 \) is an algebraic expression with a combination of constant and variable terms.
• Breaking it down, \( 3 \) is a constant term, while \(-2y\) involves both a coefficient (-2) and the variable (\( y \)).

Working with algebraic expressions means effectively rewriting them to unravel the final values of the variables involved, making the equations manageable and solvable. Understanding their structure aids in the logical processing of solving for unknowns.