Problem 15 - Algebra and Trigonometry

Question

\(9- 46\) 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 \(-7w = 15 - 2w\), we need to get all terms containing the variable \(w\) on one side. Add \(2w\) to both sides of the equation to eliminate \(-2w\) from the right-hand side. This gives: \(-7w + 2w = 15\).

Step 2: Simplify the equation

Combine like terms on the left-hand side. \(-7w + 2w\) simplifies to \(-5w\). So the equation now is: \(-5w = 15\).

Step 3: Solve for w

To find \(w\), divide both sides of the equation by \(-5\): \(w = \frac{15}{-5}\).

Step 4: Simplify the solution

The division \(\frac{15}{-5}\) simplifies to \(-3\). Thus, \(w = -3\).

Key Concepts

Solving Linear Equations

Linear equations are fundamental in algebra, consisting of variables and constants. They can be expressed in the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. These equations represent straight lines when graphed on a coordinate plane. To solve a linear equation, the goal is to find the value of the variable that makes the equation true. In the given equation \(-7w = 15 - 2w\), we aim to find the value of \(w\) that satisfies this equation. Here are some key tips for solving linear equations:

• Isolate the variable on one side of the equation to more easily solve for it.
• Use inverse operations (e.g., addition cancels subtraction) to manipulate the equation.
• Ensure any operations performed on one side are also performed on the other to maintain balance.

Successfully solving a linear equation gives us a specific solution, such as \(w = -3\), which indicates the point where this algebraic equation becomes true.

Variable Isolation

Isolating the variable is a crucial part of solving linear equations. It involves moving all the variable terms to one side of the equation. This process allows us to view the equation in its simplest form, making it easier to identify the value of the variable.Take the equation \(-7w = 15 - 2w\). The steps to isolate the variable \(w\) include:

• Add or subtract terms: We added \(2w\) to both sides, resulting in \(-7w + 2w = 15\).
• Combine like terms: Simplify the variable terms on one side. Here, \(-7w + 2w\) becomes \(-5w\).
• Final isolation: Prepare to solve for the variable by getting it alone, typically involving division or multiplication.

This systematic approach gradually turns the complicated equation into a more approachable form. Ultimately, isolation allows us to determine that \(w\) equals a specific number.

Simplifying Expressions

Simplifying expressions is an essential skill that involves reducing an equation to its most basic form without altering its value. This process makes equations easier to understand and solve. In the linear equation \(-7w + 2w = 15\), simplification takes place as follows:

• Combine like terms: On the left, we see terms involving \(w\). Combining \(-7w + 2w\) gives \(-5w\).
• Solve the result: With \(-5w = 15\), further simplification involves dividing both sides by \(-5\) to solve for \(w\).
• Final form: The expression \(\frac{15}{-5}\) simplifies to \(-3\).

These steps display the elegance of algebra, as the original equation morphs into a straightforward and manageable form. Simplification not only helps in solving the problem but also ensures the solution is concise and understandable, as seen here where \(w = -3\).