Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation algebraically. Then write the equation in the form \(f(x)=0\) and use a graphing utility to verify the algebraic solution. $$\frac{2 x}{3}+\frac{1}{2}(x-5)=6$$

Step-by-Step Solution

Answer

The solution of the given equation is \(x = \frac{51}{7}\). After rearranging the equation to \(f(x)=0\), we get \(f(x) = \frac{7x}{6} - \frac{17}{2}=0\).

Step 1: Expand the Expression

Firstly, expand the equation by distributing \(\frac{1}{2}\) into \((x-5)\), which results in \(\frac{2 x}{3}+\frac{1}{2}x-\frac{5}{2} = 6\)

Step 2: Gather and Simplify the x Terms

Next, gather the x terms together, so the equation becomes \(\frac{2 x}{3}+\frac{1}{2}x = 6+\frac{5}{2}\). Now simplify each side to obtain \(x( \frac{2}{3}+\frac{1}{2}) = \frac{17}{2}\), which further simplifies to \(\frac{7x}{6} = \frac{17}{2}\)

Step 3: Solve for x

Multiply each side by the reciprocal of \(\frac{7}{6}\), that is, \(\frac{6}{7}\), to solve for x. This gives \(x = \frac{6}{7} * \frac{17}{2} = \frac{51}{7}\)

Step 4: Set the Equation to \(f(x)=0\)

To express the equation in form \(f(x)=0\), rewrite the original equation with all terms on one side and zero on the other. This gives \(f(x)= \frac{2 x}{3} + \frac{1}{2}x - \frac{5}{2} - 6 = 0\). After simplifying, we get \(f(x) = \frac{7x}{6} - \frac{17}{2} = 0\)

Step 5: Verification

To verify the solution, substitute \(x = \frac{51}{7}\) into the equation and confirm that \(f(x)=0\). This is also confirmed when graphically represented on a graphing utility.

Key Concepts

Solving Equations

Solving algebraic equations involves finding the value(s) of the variable(s) that make the equation hold true. In the given problem, we started with the equation \( \frac{2 x}{3} + \frac{1}{2}(x-5) = 6 \).
Expanding and simplifying an equation is often necessary to isolate the variable. For instance, you distribute the \( \frac{1}{2} \) to each term within the parentheses, converting the equation to \( \frac{2 x}{3} + \frac{1}{2}x - \frac{5}{2} = 6 \).
It’s crucial to gather like terms to make the process simpler. Here, terms involving \( x \) are combined: \( \frac{2x}{3} + \frac{1}{2}x \). Then, you solve for \( x \) by performing algebraic manipulations and isolating \( x \) on one side of the equation. Multiplying both sides by the reciprocal often helps in neatly solving such fractional equations. In this problem, \( \frac{6}{7} \) was used as the reciprocal to find the value of \( x \).

Graphing Utility

Graphing utilities are tools like graphing calculators or software that allow you to visually represent equations. They can be very helpful in understanding the behavior of equations.
To use a graphing utility, first reformat your solved equation into a function form, where all expressions are set to zero, i.e., \( f(x) = \frac{7x}{6} - \frac{17}{2} = 0 \).
Entering this function into a graphing utility generates a visual graph. By analyzing this graph, you can find where the line or curve crosses the x-axis. The x-coordinate at this crossing point represents the solution of the equation. In this case, the utility should show the root \( x = \frac{51}{7} \), verifying our algebraic solution.

Verification of Solutions

Verification is the process of checking if the solution obtained is correct. It's not just about finding a solution but confirming it actually solves the given equation.
In algebra, after obtaining a solution like \( x = \frac{51}{7} \), verification includes substituting it back into the original equation: \( \frac{2 \times (51/7)}{3} + \frac{1}{2}(51/7-5) \). Ensuring that every side balances after substitution signifies correctness.
Verification can also be done graphically, as visual inspection confirms the intersection of the function with the x-axis exactly where the solution lies. Thus, the answer \( x = \frac{51}{7} \) visually corresponding to where the graph hits the horizontal axis assures us the solution is accurate.