Problem 42 - Intermediate Algebra

Question

Use a graphing calculator to solve each equation. $$ 4(x-3)-x=x-6 $$

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Simplify the equation

First, simplify the equation by expanding and combining like terms. Start by distributing the 4 across the terms inside the parentheses: \[4(x-3) = 4x - 12.\] So the equation becomes \[4x - 12 - x = x - 6.\]

Step 2: Combine like terms

Combine the 'x' terms on the left side of the equation: \[4x - x = 3x.\] This simplifies the equation to: \[3x - 12 = x - 6.\]

Step 3: Move x-terms to one side

Subtract 'x' on both sides of the equation to move all the x terms to one side: \[3x - x = x - x - 6,\] which simplifies to \[2x - 12 = -6.\]

Step 4: Isolate the x-term

Add 12 to both sides to remove the constant from the left side: \[2x - 12 + 12 = -6 + 12,\] which simplifies to \[2x = 6.\]

Step 5: Solve for x

Divide both sides by 2 to solve for x: \[x = \frac{6}{2},\] which results in \[x = 3.\]

Step 6: Verify the solution using a graphing calculator

On your graphing calculator, enter both sides of the original equation as separate functions, e.g., \(f(x) = 4(x-3) - x\) and \(g(x) = x - 6\). Observe the graph and confirm that they intersect at \(x = 3\). This verification step ensures the solution is correct.

Key Concepts

Graphing Calculator

Graphing calculators are handy tools for visualizing and solving equations, providing a graphical representation of mathematical problems. In the context of the given exercise, a graphing calculator is used to verify solutions by plotting both sides of an equation as separate functions. This means entering expressions like \(4(x-3) - x\) and \(x - 6\) into the calculator.Once both functions are entered, the calculator displays their graphs. The point where they intersect represents the solution to the equation. In this case, you'll find that they intersect at \(x = 3\), confirming the accuracy of our calculations. Using a graphing calculator not only checks your work but also helps visualize how different expressions relate graphically.

Simplifying Expressions

Simplifying mathematical expressions involves reducing them to their most compact and manageable form, often by removing parentheses and combining terms. In the original exercise, the expression \(4(x-3)\) must first be simplified. This is accomplished by distributing the 4 across the terms inside the parentheses.When you distribute a number across terms in parentheses, you multiply the number by each term within the parentheses: \[4(x-3) = 4 \cdot x - 4 \cdot 3 = 4x - 12.\]This step is crucial, as simplification transforms complex equations into forms that are easier to work with and solve. It's often the first critical step in solving linear equations.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. This requires a sequence of steps that simplify and rearrange terms. For the given equation, the process involves various maneuvers:- **Distribute and Combine Terms:** Initially, simplify the equation by distributing and combining like terms. This makes the expression as simple as possible.- **Isolate the Variable:** Move all terms involving the variable to one side. This means eliminating \(x\) from the right side by subtracting it from both sides. - **Divide and Conquer:** Finally, solve for the variable by performing arithmetic operations like division or multiplication to isolate \(x\). When we follow these steps on the example equation \(4(x-3)-x=x-6\), we eventually arrive at \(x = 3\). Each step aligns with the standard approach to solving linear equations in algebra.

Combining Like Terms

Combining like terms is an essential aspect of simplifying expressions and solving equations. Like terms are terms that contain the same variable raised to the same power. In the given exercise, you encounter like terms with the variable \(x\).For example, in the expression \(4x - x\), both terms are 'like' because they involve the same variable. Simplify by performing the arithmetic operation:- Combine: \[4x - x = 3x.\] This refinement reduces the complexity of the equation. In algebra, recognizing and combining like terms is fundamental as it consolidates expressions, making it easier to focus on the solution of the variable. The process stresses looking for terms that can blend together, streamlining the equation-solving process. This core concept simplifies the equation to a manageable form, crucial in procedure for solving linear equations.