Problem 38 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Simplify Both Sides

Let's start by simplifying both sides of the equation: \( \frac{3}{4} + \frac{1}{5}x - \frac{1}{2} = \frac{4}{5}x \).First, simplify the left-hand side:- Combine the fractions \( \frac{3}{4} - \frac{1}{2} \). To subtract these, use a common denominator of 4: \[ \frac{3}{4} - \frac{1}{2} = \frac{3}{4} - \frac{2}{4} = \frac{1}{4} \].So the equation becomes: \( \frac{1}{4} + \frac{1}{5}x = \frac{4}{5}x \).

Step 2: Isolate x

Next, we want to isolate \(x\):Subtract \(\frac{1}{5}x\) from both sides:\[\frac{1}{4} = \frac{4}{5}x - \frac{1}{5}x\]This simplifies to:\[\frac{1}{4} = \frac{3}{5}x\]

Step 3: Solve for x

Now, solve for \( x \) by multiplying both sides by the reciprocal of \( \frac{3}{5} \), which is \( \frac{5}{3} \):\[x = \frac{1}{4} \times \frac{5}{3} = \frac{5}{12}\].Thus, \( x = \frac{5}{12} \).

Step 4: Verify the Solution

To verify, substitute \( x = \frac{5}{12} \) back into the original equation:\[\frac{3}{4} + \frac{1}{5} \times \frac{5}{12} - \frac{1}{2} = \frac{4}{5} \times \frac{5}{12}\]Calculating each term:- \( \frac{1}{5} \times \frac{5}{12} = \frac{1}{12} \)- LHS: \( \frac{3}{4} + \frac{1}{12} - \frac{1}{2} = \frac{1}{12} \) (after common denominator)- RHS: \( \frac{4}{5} \times \frac{5}{12} = \frac{1}{3} \) (now, further simplifying it to also \(\frac{1}{12}\))The left side equals the right side, verifying our solution.

Step 5: Graphical Support

To support this solution graphically:- Plot \( y = \frac{1}{4} + \frac{1}{5}x \) and \( y = \frac{4}{5}x \) on the same graph.- The point of intersection is where the solutions hold true.- The graphs should intersect at \( x = \frac{5}{12} \), which confirms the solution.

Key Concepts

Analytical Solutions

An analytical solution involves solving equations using algebraic manipulations. The goal is to isolate the variable on one side of the equation. In our exercise, we start with the equation:\[\frac{3}{4} + \frac{1}{5}x - \frac{1}{2} = \frac{4}{5}x\]Our first step is to simplify fractions on one side. By finding a common denominator, we combine \( \frac{3}{4} \) and \( -\frac{1}{2} \), resulting in \( \frac{1}{4} \). This gives us:\[\frac{1}{4} + \frac{1}{5}x = \frac{4}{5}x\]Next, we isolate \( x \) by subtracting \( \frac{1}{5}x \) from both sides and simplifying further, leading to:\[\frac{1}{4} = \frac{3}{5}x\]To find \( x \), multiply both sides by the reciprocal of \( \frac{3}{5} \), giving us:\[x = \frac{5}{12}\]This approach shows how we can use algebraic techniques to solve equations without the need for guesswork.

Graphical Verification

Graphical verification helps confirm the solution we find analytically by visualizing it. By plotting the expressions as graphs, we can see where the two sides of the equation intersect. In this case, we graph:

• \( y = \frac{1}{4} + \frac{1}{5}x \)
• \( y = \frac{4}{5}x \)

The point where these two lines intersect is the solution to the equation. After plotting them, we see that they meet at \( x = \frac{5}{12} \). This intersection confirms that our analytical solution is correct.
Graphical verification is an excellent way to double-check our work, as a visual representation can enhance understanding and ensure accuracy.

Fraction Operations

Performing fraction operations correctly is crucial in algebra, especially when variables are involved. In our exercise, we started by simplifying complex fractions:

• Determine a common denominator for fractions on the same side, like converting \( \frac{1}{2} \) to \( \frac{2}{4} \) so it can be subtracted from \( \frac{3}{4} \).


• Combine fractions by adding or subtracting the numerators, keeping the denominator the same.

Once simplified, operations such as adding and subtracting like terms help to isolate the unknown variable. Finally, solving the equation may involve multiplying by the reciprocal of a fraction, as seen when we solved for \( x \), resulting in:\[x = \frac{1}{4} \times \frac{5}{3} = \frac{5}{12}\]Having a strong grasp of fraction operations makes solving algebraic equations much smoother and more accurate.