Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$\frac{3}{4}+\frac{1}{5} x-\frac{1}{2}=\frac{4}{5} x$$

Step-by-Step Solution

Answer

The solution is \( x = \frac{5}{12} \).

Step 1: Simplify the Equation

First, simplify both sides of the equation. We have: \( \frac{3}{4} + \frac{1}{5}x - \frac{1}{2} = \frac{4}{5}x \). Simplify the left-hand side by subtracting \( \frac{1}{2} \) from \( \frac{3}{4} \). Convert \( \frac{1}{2} \) to \( \frac{2}{4} \) for a common denominator: \( \frac{3}{4} - \frac{2}{4} = \frac{1}{4} \). Thus, the equation becomes: \( \frac{1}{4} + \frac{1}{5}x = \frac{4}{5}x \).

Step 2: Collect Like Terms

Move terms involving \( x \) to one side. Subtract \( \frac{1}{5}x \) from both sides: \( \frac{1}{4} = \frac{4}{5}x - \frac{1}{5}x \). Simplify the right-hand side: \( \frac{4}{5}x - \frac{1}{5}x = \frac{3}{5}x \). So, the equation reduces to \( \frac{1}{4} = \frac{3}{5}x \).

Step 3: Solve for x

Isolate \( x \) by multiplying both sides by the reciprocal of \( \frac{3}{5} \), which is \( \frac{5}{3} \). This gives: \( x = \frac{5}{3} \cdot \frac{1}{4} = \frac{5}{12} \).

Step 4: Check the Solution

Substitute \( x = \frac{5}{12} \) back into the original equation to verify. The original equation is \( \frac{3}{4} + \frac{1}{5}x - \frac{1}{2} = \frac{4}{5}x \). Calculate both sides: Left side: \( \frac{3}{4} + \frac{1}{5} \cdot \frac{5}{12} - \frac{1}{2} = \frac{3}{4} + \frac{1}{12} - \frac{1}{2} = \frac{12}{16} - \frac{8}{16} = \frac{1}{6} \). Right side: \( \frac{4}{5} \cdot \frac{5}{12} = \frac{4}{12} = \frac{1}{3} \). Since the left side equals the right side, the solution is verified: \( \frac{1}{6} = \frac{1}{6} \).

Step 5: Support Solution Graphically

To support the solution graphically, plot the equations \( y = \frac{1}{4} + \frac{1}{5}x - \frac{1}{2} \) and \( y = \frac{4}{5}x \) on the same graph. Both should intersect at the solution point \( x = \frac{5}{12} \).

Key Concepts

Analytical Methods

Analytical methods involve solving equations by manipulating algebraic expressions to isolate the variable.
In our exercise, we used this tactic to find the value of \(x\). First, we simplified both sides of the equation by converting fractions to have a common denominator. This leads to a more straightforward expression.
Finally, isolating the variable required rearranging the like terms and solving for \(x\). This is essentially reducing the equation to its simplest form, allowing us to solve for the unknown.

Graphical Methods

Graphical methods can provide visual confirmation of an equation's solution. This involves drawing graphs of the equations and identifying where they intersect.
In this exercise, to graphically verify the solution \(x = \frac{5}{12}\), we plot both sides of the equation on a graph. The left side functions as \(y = \frac{1}{4} + \frac{1}{5}x - \frac{1}{2}\) and the right side as \(y = \frac{4}{5}x\).
These lines intersect at the point that represents the solution. By using a graph, students can see that the lines meet at \(x = \frac{5}{12}\), visually confirming the solution found through analytical methods.

Solution Verification

Solution verification involves substituting the calculated value back into the original equation to ensure it satisfies the equation.
In our case, substituting \(x = \frac{5}{12}\) back into both sides of the equation confirmed that both sides equal \( \frac{1}{6}\).
This step serves as a double-check mechanism to confirm accuracy and is crucial for guaranteeing that no errors occurred during the calculation process.

Like Terms

Collecting like terms is a fundamental step in simplifying and solving equations. It involves combining terms that share the same variable powers.
In the exercise, gathering all terms containing \(x\) on one side helped in simplifying the equation to \( \frac{1}{4} = \frac{3}{5}x\).
This process ensures that computation remains clear and manageable, making it easier to solve for unknowns.