Problem 80 - Prealgebra

Question

The problems below review some of the material on solving equations. Reviewing these problems will help you with the next section. Solve. $$7 x-3=5 x+9$$

Step-by-Step Solution

Answer

The solution is \(x = 6\).

Step 1: Move Variable Terms to One Side

First, we need to isolate the variable terms on one side of the equation. We can do this by subtracting \(5x\) from both sides of the equation: \[7x - 5x - 3 = 5x - 5x + 9\] This simplifies to: \[2x - 3 = 9\]

Step 2: Move Constant Term to Other Side

Now, we need to isolate the term with the variable \(x\). Add \(3\) to both sides of the equation to move the constant term:\[2x - 3 + 3 = 9 + 3\] This simplifies to: \[2x = 12\]

Step 3: Solve for x

Now, we have to solve for \(x\). Divide both sides of the equation by \(2\) to get the value of \(x\): \[\frac{2x}{2} = \frac{12}{2}\] This simplifies to: \[x = 6\]

Key Concepts

Isolating Variable

When solving linear equations, the goal is to determine the value of the unknown variable. One common strategy is to isolate the variable on one side of the equation. This makes it easier to understand what the variable represents. Starting with the equation \(7x - 3 = 5x + 9\), we can isolate the variable by first getting all terms containing \(x\) on one side. We do this by subtracting \(5x\) from both sides, effectively moving all the \(x\) terms to one location. This manipulation results in \(2x - 3 = 9\).

Isolating the variable further involves removing any additional constants or numbers attached to the \(x\) term. By adding \(3\) to both sides next, you simplify the equation to \(2x = 12\). As a result, the equation becomes much easier to solve, with \(x\) almost on its own, leading to the final step.

• Move variable terms to one side.
• Move constants away from \(x\).
• Simplify the expression.

Equation Simplification

Equation simplification is a critical step when working with linear equations, turning complex expressions into simpler ones. By performing operations like addition, subtraction, multiplication, or division, you can alter the form of an equation without changing its equality. This step is vital to making the problem more manageable.

In the original problem, once you subtract \(5x\) from both sides, the equation \(7x - 3 = 5x + 9\) changes to \(2x - 3 = 9\). Subtracting \(5x\) simplifies the expression by removing the extra \(x\) terms. Following this, adding \(3\) to both sides changes the expression further to \(2x = 12\), removing the constants floating with \(x\). Such simplifications help in clearing unnecessary terms, making the path to solving for \(x\) much clearer.

• Identify opportunities for simplification.
• Use basic arithmetic operations.
• Simplify to an equation with only the variable and its coefficient.

Mathematical Problem Solving

Mathematical problem solving involves several strategies and methods to find solutions effectively. With linear equations, such as \(7x - 3 = 5x + 9\), following each step in order lets us reach a solution. Initially, focus on isolating the variable by rearranging the equation, which is a crucial step.

After simplification, the final step in problem solving is solving for the variable, which we achieve by dividing both sides of \(2x = 12\) by \(2\). This step reveals that \(x = 6\). Ultimately, mathematical problem solving takes a systematic approach.

Writing down each step helps organize your thoughts and ensures accuracy. Try to verify your solution by plugging the value back into the original equation. If it satisfies both sides, the solution is likely correct. Following this systematic approach builds strong problem-solving skills that can be applied to more advanced mathematics later on.

• Follow each step in sequence.
• Break complex problems into simple tasks.
• Verify solutions through back-checking.