Problem 26 - Elementary Algebra

Question

. \(-2 x-y=9\) for \(y\)

Step-by-Step Solution

Answer

The solution is \(y = -2x - 9\).

Step 1: Identify the Equation to Solve

The equation given is \(-2x - y = 9\). Our goal is to solve for \(y\), meaning we want to express \(y\) in terms of \(x\).

Step 2: Isolate y by Adding Terms

Add \(2x\) to both sides of the equation to move the \(x\)-related term to the right side. The equation becomes \(-y = 2x + 9\).

Step 3: Solve for y by Changing Sign

To get \(y\) by itself, multiply each term in the equation \(-y = 2x + 9\) by \(-1\). This simplifies to \(y = -2x - 9\).

Key Concepts

Solving Equations

Solving equations is a fundamental concept in algebra that involves finding unknown values. It is like solving a puzzle where the goal is to discover what number makes the equation true. When dealing with equations, you are often given a mathematical statement involving an equal sign, dividing the statement into two parts. The aim is to determine the value(s) of the unknown variable(s) that satisfy the equation.

To solve equations, you can perform the same operation on both sides of the equation without changing its equality. This might include:

• Adding or subtracting the same number from both sides
• Multiplying or dividing both sides by the same non-zero number
• Re-arranging the terms

By carefully applying these operations, you gradually simplify the equation, making it easier to solve for the unknown variable.

Linear Equations

Linear equations are a specific type of equation where each term is either a constant or the product of a constant and a single variable. These equations form straight lines when graphed and have no variables raised to a power higher than one.

For example, the equation \(-2x - y = 9\) is a linear equation because it consists of variables which are only to the first power. Linear equations can be expressed in different forms, such as the standard form \(Ax + By = C\) or the slope-intercept form \(y = mx + b\). Here, \(-y = 2x + 9\) was rearranged for ease of solving to its slope-intercept form \(y = -2x - 9\). This form explicitly shows the slope and y-intercept, allowing for easy graphing and interpretation.

Isolation of Variables

Isolation of variables is a technique used in algebra to solve equations by getting the variable of interest alone on one side of the equation. This is a critical step in the process of solving equations as it helps reveal the value of the unknown.

In the original exercise, the purpose was to isolate \(y\) in the equation \(-2x - y = 9\). By manipulating the equation using algebraic operations, we successfully isolated \(y\).

Here’s how it was done:

• First, add \(2x\) to both sides to start moving terms with \(x\) away from \(y\).
• Now you have \(-y = 2x + 9\).
• Multiply the entire equation by \(-1\) to change the sign of \(y\), which results in \(y = -2x - 9\).

By following these steps, you systematically isolated \(y\), making it the subject of the equation and thereby solving for \(y\) in terms of \(x\).