Problem 60 - Algebra A Combined Function

Question

Solve. See Examples 1 through 7 $$ 3-\frac{1}{2} x=5 x-8 $$

Step-by-Step Solution

Answer

The solution is \(x = 2\).

Step 1: Move terms involving x to one side

To begin solving the equation \(3 - \frac{1}{2}x = 5x - 8\), we need to collect all the \(x\)-terms on one side. Add \(\frac{1}{2}x\) to both sides:\[3 = 5x + \frac{1}{2}x - 8\]This simplifies to:\[3 = (5 + \frac{1}{2})x - 8\]

Step 2: Simplify and move constants to the other side

Simplify \(5 + \frac{1}{2}\) which is equivalent to \(\frac{11}{2}\), then the equation becomes:\[3 = \frac{11}{2}x - 8\]Next, add 8 to both sides to move the constant term:\[3 + 8 = \frac{11}{2}x\]Thus, \(11 = \frac{11}{2}x\).

Step 3: Solve for x

To solve for \(x\), multiply both sides by \(\frac{2}{11}\) to isolate \(x\):\[11 \times \frac{2}{11} = \frac{11}{2}x \times \frac{2}{11}\]\[x = 2\]

Step 4: Verify the solution

Substitute \(x = 2\) back into the original equation:\[3 - \frac{1}{2}(2) = 5(2) - 8\]Simplify both sides:\[3 - 1 = 10 - 8\]Both sides equal 2, confirming that \(x = 2\) is the correct solution.

Key Concepts

Linear Equations

Linear equations are a type of equation where the variable is raised only to the power of 1. These equations usually look like ax + b = c, where a, b, and c are constants. Linear equations can be graphed as straight lines on a coordinate plane. Understanding linear equations is essential because they form the foundation for more complex algebraic concepts. Linear equations can have:

• One solution, where the graph intersects the x-axis at one point.
• No solution, which occurs if the lines are parallel.
• Infinite solutions, when the equations represented are the same line.

This predictability makes linear equations a manageable starting point for learning algebra.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. To do this:

• Identify the variable you need to solve for.
• Perform operations on both sides of the equation to simplify and isolate the variable. This maintains the equality.
• Remember that whatever you apply to one side, you must apply to the other.

By practicing these steps, you become skilled at solving linear equations efficiently and accurately. Don't forget to keep your work organized! This will help you avoid errors that can occur by skipping steps or losing track of terms.

Variable Isolation

Variable isolation is part of solving equations where we try to get the variable alone on one side. The main goal is to make our equation look like x = value. Here are methods to isolate the variable:

• Add or subtract terms from both sides to remove any constants from the side with the variable.
• Once you have the variable and a coefficient, divide or multiply to eliminate the coefficient.
• Simplify each step to avoid mistakes and make the process smoother.

In the example provided, we started by moving all x terms to one side and constants to the other. As you practice more, you'll find isolating variables becomes second nature.

Verification of Solutions

Verification ensures that your solution is correct. After solving an equation, substitute the solution back into the original equation. This confirms that both sides equal when the variable takes on this solution. Let's check why verification is crucial:

• Prevents errors by double-checking your solution.
• Gives confidence in the accuracy of your algebraic operations.
• Serves as a way to validate your approach and reasoning.

For instance, in the provided example, substituting x = 2 into both sides of the original equation confirmed the solution, thus ensuring its correctness.