Problem 19 - Algebra and Trigonometry

Question

The given equation is either linear or equivalent to a linear equation. Solve the equation. \(-7 w=15-2 w\)

Step-by-Step Solution

Answer

The solution is \(w = -3\).

Step 1: Move Variables to One Side

To solve the equation, start by moving all terms involving the variable to one side. Add \(2w\) to both sides to get: \(-7w + 2w = 15\).

Step 2: Simplify the Equation

Combine like terms on the left side: \(-7w + 2w = -5w\). Now the equation becomes \(-5w = 15\).

Step 3: Solve for the Variable

To isolate \(w\), divide both sides by \(-5\): \(w = \frac{15}{-5}\). This simplifies to \(w = -3\).

Key Concepts

Algebra Concepts

Algebra is a fundamental branch of mathematics focused on solving equations and understanding relationships between variables. It is essential for analyzing mathematical expressions and discovering unknown values.
One key aspect of algebra is the concept of variables, which are symbols representing unknown numbers that we aim to determine. By integrating rules and operations like addition, subtraction, multiplication, and division—typically learned in basic arithmetic—algebra allows us to explore and solve problems logically.

• Variables: Symbols (often letters such as x, y, w) that denote unknown numbers.
• Simplification: The process of making an equation easier to understand and solve.
• Operations: Addition, subtraction, multiplication, and division used to solve equations.

Understanding these foundational concepts enables students to handle more complex mathematical problems efficiently.

Linear Equations

Linear equations are equations that graph as straight lines, signaling a direct relationship between the variables involved. They're characterized by their constant rates of change and the absence of exponents greater than one on the variable.
Linear equations come in several forms, such as the slope-intercept form \(y = mx + b\), but they often need to be manipulated to solve for a specific variable, as seen in the exercise

• Simplicity: Linear equations are straightforward and involve adding, subtracting, multiplying, or dividing terms.
• Structure: Their general form is \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.
• Solution: Solving typically involves isolating the variable to discover its value.

Mastering linear equations is vital, as it lays the groundwork for more advanced topics like linear inequality and proportional reasoning.

Variable Isolation

The process of isolating a variable is a core strategy in solving equations, which involves rearranging the equation to have the variable alone on one side of the equation.
This concept is crucial not only in linear equations but in almost all algebraic expressions.
To isolate the variable, perform inverse operations to "undo" whatever mathematical operations have been applied to it. For example:

• In the equation \(-7w = 15 - 2w\), adding \(2w\) to both sides helps to move all the terms involving \(w\) to one side, simplifying the equation.
• Following this, combine like terms to further simplify it.
• Lastly, divide both sides by the coefficient of \(w\) to get \(w = -3\).

Effectively mastering variable isolation will assist students in tackling more complex algebraic problems with confidence.