Problem 39 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$5 x-(8-x)=2[-4-(3+5 x-13)]$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{5}{4}\).

Step 1: Simplify the Equation

First, simplify both sides of the equation. Start with the left side: \[5x - (8 - x) = 5x - 8 + x = 6x - 8.\]Now simplify the right side, distributing the 2 through the bracket: \[2[-4 - (3 + 5x - 13)] = 2[-4 - 3 - 5x + 13] = 2[6 - 5x] = 12 - 10x.\] The equation now is: \[6x - 8 = 12 - 10x.\]

Step 2: Combine Like Terms

Add \(10x\) to both sides to bring all \(x\)-terms to one side of the equation:\[6x + 10x - 8 = 12.\]Combine the \(x\)-terms:\[16x - 8 = 12.\]

Step 3: Isolate the Variable

Add 8 to both sides to remove the constant term on the left:\[16x = 20.\]

Step 4: Solve for x

Divide both sides by 16 to solve for \(x\):\[x = \frac{20}{16} = \frac{5}{4}.\]

Step 5: Check the Solution

Substitute \(x = \frac{5}{4}\) back into the original equation to verify:Left side: \[5 \left(\frac{5}{4}\right) - \left(8 - \frac{5}{4}\right) = \frac{25}{4} - \left(\frac{32}{4} - \frac{5}{4}\right) = \frac{25}{4} - \frac{27}{4} = -\frac{2}{4} = -\frac{1}{2}.\] Right side: \[2[-4-(3+5(\frac{5}{4})-13)] = 2[-4 - \left(3 + \frac{25}{4} - 13\right)] = 2[-4 - \left(\frac{12}{4} + \frac{25}{4} - \frac{52}{4}\right)] = 2[-4 - (-\frac{15}{4})] = 2[\frac{1}{4}] = \frac{1}{2}.\]Both sides are equal, confirming the solution.

Step 6: Graphical Representation

Graph both sides of the equation as functions of \(x\) to visually verify the solution. Plot \(y_1 = 6x - 8\) and \(y_2 = 12 - 10x\) and find the intersection point.The intersection point should occur at \(x = \frac{5}{4}\), confirming the analytic solution.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is the fundamental process of rearranging and simplifying equations. This is often the first step in solving most algebraic problems.
In the given equation, we start by simplifying each side individually. Let's break down a few processes to comprehend this better:

• **Distributive Property**: To eliminate parentheses, we distribute the numbers across each term inside the brackets. For example, the right side of the equation includes the expression \(2[-4 - (3 + 5x - 13)]\). When simplified, it becomes \(2[6 - 5x]\).
• **Combining Like Terms**: Next, we combine the similar terms from each side. The left side \(5x - (8 - x)\) transforms into \(6x - 8\) after combining similar terms. Similarly, simplifying the right side gives us \(12 - 10x\).

These manipulations reduce the equation to a simpler form, ready for solving.

Variable Isolation

Variable isolation is a key technique when solving equations. It involves moving all terms involving the variable to one side of the equation.
This process ensures that the variable (here \(x\)) is isolated, simplifying the solution:

• **Add/Subtract Constants**: In the equation \(6x - 8 = 12 - 10x\), the first step is to add \(10x\) to both sides to gather all \(x\) terms on one side. This results in \(16x - 8 = 12\).
• **Solving for Variable**: Once \(x\) terms are isolated, we eliminate any constants deterring the isolation of the variable. Adding \(8\) to both sides gives \(16x = 20\).
• **Divide Coefficient**: Finally, divide both sides by \(16\) to solve for \(x\), resulting in \(x = \frac{5}{4}\).

Through these steps, the variable \(x\) is effectively isolated, leading us directly to the solution.

Graphical Verification

Graphical verification involves plotting equations as functions of their variables and finding their intersection points. This step provides a visual confirmation of the algebraic solution.
Here's how it works in practice for our equation:

• **Define Functions**: Represent both sides of the equation as functions of \(x\). In our example, \(y_1 = 6x - 8\) and \(y_2 = 12 - 10x\).
• **Plot the Graphs**: On a graph panel, plot \(y_1\) and \(y_2\). Use a suitable range that encompasses the calculated solution \(x = \frac{5}{4}\).
• **Determine Intersection**: The point where both graphs intersect corresponds to the solution of the equation. Confirm that the intersection occurs at \(x = \frac{5}{4}\), validating the solution found analytically.

Graphical verification helps in substantiating algebraic results through visual means, providing a deeper understanding of equation solutions.