Problem 61 - Elementary Algebra

Question

For Problems 55-70, solve each equation for the indicated variable. (Objective 4) $$ y=-3 x-4 \quad \text { for } x $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{y + 4}{-3} \).

Step 1: Identify the Equation

The given equation is \( y = -3x - 4 \) and we need to solve for \( x \).

Step 2: Isolate the Linear Term

Add 4 to both sides of the equation to isolate the terms containing \( x \) on the right side.\[y + 4 = -3x - 4 + 4\]This simplifies to:\[y + 4 = -3x\]

Step 3: Solve for x

Divide each side by -3 to solve for \( x \):\[x = \frac{y + 4}{-3}\]

Key Concepts

Isolating Variables

When solving linear equations, it's crucial to know how to isolate variables. Isolating the variable means getting the variable of interest by itself on one side of the equation. This process is fundamental because it allows you to solve for that variable explicitly.
In our example, we have the equation: \( y = -3x - 4 \). We want to solve for \( x \), which means \( x \) needs to be by itself on one side of the equation.
Here’s how it goes:

• First, you identify what operations are currently being applied to \( x \). In this case, \( x \) is being multiplied by -3 and then subtracted by 4.
• To isolate \( x \), you need to perform inverse operations. These are steps that reverse what’s currently being done to the variable.
• Start by eliminating any constants added or subtracted, which in this example, means adding 4 to both sides.
• The result becomes: \( y + 4 = -3x \), which is one step closer to isolating \( x \).

Remember, whatever you do to one side of the equation, you must do to the other. This keeps the equation balanced and ensures that your solution remains correct.

Linear Terms

In algebra, linear terms are expressions where the variable has an exponent of one. They are straightforward, having the form \( ax + b \), where \( a \) and \( b \) are constants. In our exercise, the linear term is \(-3x\).
This term describes a straight line when graphed. Therefore, solving linear equations often involves dealing with these types of terms.

• Such expressions are significant because they are simple and easy to manipulate.
• Understanding linear terms helps in identifying the easiest path to isolate the variable you are solving for. In this exercise, isolating \( x \) involves managing the linear term \(-3x\).
• Notice that the coefficient of the linear term, which is \(-3\) here, plays a crucial role in scaling the variable. Thus, we must deal with it to properly isolate and solve for the variable of interest.

Since the linear term directly influences the variable, managing it correctly is key to finding the solution.

Equation Manipulation

Equation manipulation involves using mathematical operations to rearrange equations effectively. This is particularly essential when solving for a specific variable, as it often requires moving terms around to simplify the given equation.
In the provided example, equation manipulation is employed to solve for \( x \). The steps include:

• Adding or subtracting terms to both sides to change the structure of the equation. In our case, we added 4 to move constant terms away from the \( x \)-term.
• Once similar terms are collected (getting \( y + 4 = -3x \)), the next phase is changing the coefficient of \( x \) to 1 through division. Here, we divide every term by \(-3\).
• The final expression shows \( x \) isolated: \( x = \frac{y + 4}{-3} \).

This kind of manipulation requires practice to become intuitive. Key strategies include recognizing when to use addition, subtraction, multiplication, or division to rearrange parts of the equation efficiently.
By mastering these manipulation tactics, solving complex equations becomes a much more manageable task.