Problem 39 - Algebra for College Students

Question

Solve each equation for the indicated variable. \(-7 x-y=4\) for \(y\)

Step-by-Step Solution

Answer

y = -7x - 4

Step 1: Identify the equation

The given equation is \(-7x - y = 4\).We need to solve this equation for the variable \(y\).

Step 2: Isolate the term with y

To isolate \(y\), add \(-7x\) to both sides of the equation:\(-y = 7x + 4\).

Step 3: Solve for the variable y

To solve for \(y\), multiply both sides of the equation by \(-1\):\(y = -7x - 4\).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra. It involves finding the value of a variable that makes an equation true. In this guide, we are focusing on linear equations which include variables raised to the first power. These types of equations are particularly accessible and offer a great introduction to equation solving. Solving any equation generally involves three steps:

• Understanding the problem by identifying the given equation.
• Manipulating the equation to progressively find the desired variable.
• Ensuring that each mathematical operation maintains the balance of the equation.

Remember, equations are like scales. What you do to one side, you must do to the other to keep them balanced. This principle is crucial as you work towards isolating and finding the solution for the unknown variable.

Isolation of Variables

Isolation of variables is the key process in solving algebraic equations. This refers to the method of rearranging an equation so that one variable stands alone on one side of the equation.For instance, consider the equation \(-7x - y = 4\).We aim to solve for \(y\), which means we need \(y\) by itself on one side of the equality.To achieve this:

• Identify the term containing the variable you want to isolate, in this case, \(-y\).
• Use algebraic operations to move all other terms to the opposite side. Here, we add \(-7x\) to both sides, resulting in \(-y = 7x + 4\).
• This step simplifies our target variable \(-y\), now to isolate \(y\) completely, multiply both sides by \(-1\) yielding \(y = -7x - 4\).

This isolation allows us to express the equation in terms of \(y\) clearly, setting the foundation for solving for any given input of \(x\).

Algebraic Manipulation

Algebraic manipulation involves the methodical application of mathematical operations to rearrange equations. This process ensures that you efficiently and accurately isolate variables. In the context of our example, understanding algebraic manipulation is vital.To manipulate the equation \(-7x - y = 4\):

• Recognize the operations required to isolate the target variable, \(-y\).
• We begin by adding \(-7x\) to eliminate it from the left side, resulting in \(-y = 7x + 4\).
• The next step involves eliminating the negative sign in front of \(y\) by multiplying the entire equation by \(-1\). This changes all signs, giving \(y = -7x - 4\).

Algebraic manipulation requires careful selection of operations, ensuring each step logically follows the previous. This meticulous process is central to solving equations effectively, forming a solid base for more complex algebraic challenges.