Problem 8 - Elementary Algebra

Question

Solve. $$ -5 x+3=-8 x-9 $$

Step-by-Step Solution

Answer

The solution is \(x = -4\).

Step 1: Eliminate the Variable on One Side

Begin by adding \(8x\) to both sides of the equation to eliminate \(-8x\) from the right side. This will give: \[-5x + 8x + 3 = -9\].

Step 2: Simplify the Equation

Combine the like terms on the left side of the equation. \[(8x - 5x) + 3 = -9\] simplifies to \[3x + 3 = -9\].

Step 3: Isolate the Variable Term

Subtract 3 from both sides to isolate the term with \(x\). This gives us: \[3x = -9 - 3\].

Step 4: Solve for the Variable

Simplify the right side of the equation to further isolate \(x\). \[3x = -12\] Now, divide both sides by 3: \[x = \frac{-12}{3}\].

Step 5: Simplify the Value of the Variable

Divide \(-12\) by \(3\) to find the value of \(x\). \[x = -4\].

Key Concepts

Solving Equations

Solving equations is the process of finding the values of variables that make the equation true. It involves using mathematical operations to manipulate the equation to find the specific value or values of the unknowns. The primary goal is to "solve" or find the variable's value by performing operations that maintain the balance of the equation.
Solving involves:

• Recognizing what needs to be solved for.
• Determining the steps necessary to isolate the variable.
• Using mathematical operations strategically.

This ensures that your solution is both accurate and efficient. Each step in solving an equation should bring you closer to finding the value of the unknown variable.

Variable Isolation

Variable isolation is the technique of rearranging an equation to get the unknown variable by itself on one side of the equation. This is crucial for solving linear equations since it allows you to identify the exact value of the variable.

To isolate a variable:

• Move all terms containing the variable to one side of the equation.
• Eliminate any constants or coefficients attached to the variable by using inverse operations (like addition if there's subtraction, or division if there's multiplication).

The main idea is to simplify the equation gradually until the variable is isolated, making it easier to solve for its value.

Combining Like Terms

Combining like terms is an essential part of simplifying algebraic expressions, including linear equations. It involves combining terms that have the same variable raised to the same power. This step helps to consolidate the equation into a simpler form.

To combine like terms:

• Identify terms that have identical variable parts.
• Add or subtract their coefficients only, keeping the variable part unchanged.

For instance, in the intermediate step of our solution, combining \(8x - 5x\) resulted in \(3x\). Simplifying the equation by combining like terms makes it easier to manipulate and eventually solve the equation.

One-Step Equations

One-step equations are the simplest form of linear equations where the solution can be found in a single operation. These types of equations have been reduced to their simplest form, allowing you to solve them in just one calculation.

Solving one-step equations involves performing the inverse operation of the final operation applied to the variable. For example:

• If the variable is added to something, subtract that same thing from both sides.
• If the variable is multiplied by a number, divide both sides by that number.

In our exercise, once we reached \(3x = -12\), we divided both sides by 3 to solve for \(x\). This highlights the simplicity and efficiency of solving one-step equations once they are isolated.