Problem 28 - Elementary Algebra

Question

. \(-x-y=9\) for \(x\)

Step-by-Step Solution

Answer

Solve for x: \(x = -y - 9\).

Step 1: Understand the Equation Structure

The equation given is \(-x - y = 9\). This is a linear equation with two variables, \(x\) and \(y\). Our goal is to solve for \(x\), which means expressing \(x\) in terms of \(y\).

Step 2: Isolate the Variable \(x\)

To solve for \(x\), you need to isolate \(x\) on one side of the equation. Start by adding \(x\) to both sides of the equation: \(x - y - x = x + 9\), simplifying to \(-y = x + 9\).

Step 3: Express \(x\) in terms of \(y\)

Continue isolating \(x\) by moving 9 to the other side of the equation: \(-y - 9 = x\). This can be rewritten as \(x = -y - 9\).

Step 4: Double Check Your Work

Verify your work by plugging a value for \(y\) into the expression and ensuring the original equation holds. This reinforces the solution's correctness.

Key Concepts

Linear Equations

A linear equation is an algebraic expression where each term is either a constant or the product of a constant and a single variable.
Linear equations can be written in the form \(ax + b = c\), where each term can contain constants and variables.
In the provided original exercise, we have two variables, \(x\) and \(y\), represented in the equation \(-x - y = 9\).
This equation is linear because the highest power of both variables is 1.
Linear equations can model real-world situations, helping us to find unknowns in various scenarios.
Understanding linear equations gives you the tools to tackle more complex algebraic problems with ease.
When dealing with linear equations:

• Remember that they graph as straight lines.
• Expect only one solution when solving for one variable if no restrictions are added.
• Be prepared to rearrange terms to isolate the desired variable, as shown in the given solution.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to rearrange and simplify equations.
This skill is crucial for isolating variables, especially when dealing with linear equations.
In solving the equation \(-x - y = 9\), algebraic manipulation was used to isolate \(x\).
Here’s a breakdown of key steps taken in the solution:

• First, \(-x\) is added to both sides to help reposition \(x\) on one side of the equation.
• This results in a transition to \(-y = x + 9\).
• Algebraic manipulation is further used by subtracting \(9\) from both sides, leading to \(x = -y - 9\).

Each manipulation maintains the equality, ensuring that the solution remains valid.
Knowing how to rearrange terms efficiently can simplify finding solutions in algebra.

Two-Variable Equations

Two-variable equations involve working with two unknowns, often represented as \(x\) and \(y\).
These types of equations can form a plane when graphed, allowing you to see the relationship between the variables visually.
In the exercise, the equation \(-x - y = 9\) is an example of a two-variable equation where we are tasked with solving for one variable in terms of the other.
When dealing with two-variable equations:

• Understand that solutions may not be single values, but pairs that satisfy both variables.
• You often need to express one variable in terms of another, as was necessary in this problem.
• Such equations frequently lead to systems of equations, where more than one equation is solved simultaneously.

Grasping two-variable equations equips you to handle more complex algebraic problems where multiple relationships need exploration.
This forms a foundation for studying functions and interpreting graphs in math.