Problem 37 - 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. $$\frac{x-5}{4}+\frac{x}{2}=10$$

Step-by-Step Solution

Answer

The solution to the equation \(\frac{x-5}{4}+\frac{x}{2}=10\) is \(x=15\). The equation in the form \(f(x)=0\) is \(f(x)=\frac{x-5}{4}+\frac{x}{2}-10\). This is verified visually with a graph, with an intersection at \(x=15\).

Step 1: Simplify the Equation

To simplify the equation \(\frac{x-5}{4}+\frac{x}{2}=10\), start by finding a common denominator, which is 4. Thus, your equation becomes \(\frac{x-5}{4}+\frac{2x}{4}=10\). Combining like terms gives \(\frac{3x-5}{4}=10\).

Step 2: Solve the Equation

After simplification, you can now solve for x. Start by multiplying every term by 4 to get rid of the denominator: \(3x-5=40\). Then, add 5 to both sides of the equation, so you end up with \(3x=45\). Finally, divide both sides by 3 to isolate x, leading to \(x=15\) as the solution.

Step 3: Arrange the Equation in the form \(f(x)=0\)

Rewrite our original equation as \(f(x)=\frac{x-5}{4}+\frac{x}{2}-10\). Then putting \(x=15\) into \(f(x)\), it gives \(f(15)=0\), which verifies that the solution \(x=15\) is correct.

Step 4: Use a Graphing Utility to Verify

Plot the function \(f(x)=\frac{x-5}{4}+\frac{x}{2}-10\) using a graphing utility. The x-coordinate of the point where the graph intersects the x-axis is the solution to the equation \(f(x)=0\). In this case, you should find the graph intersects the x-axis at \(x=15\), further verifying the correctness of the solution obtained.

Key Concepts

Common Denominator

When solving equations that include fractions, finding a common denominator is crucial. This allows you to combine fractions into a simpler expression. Suppose you want to solve the equation \( \frac{x-5}{4}+\frac{x}{2}=10 \). The first step involves identifying the least common denominator (LCD) among the fractions involved. Here, the denominators are 4 and 2.

Notice that 4 is a multiple of 2, so 4 can serve as our LCD. Express each fraction using this common denominator:

• First term: \( \frac{x-5}{4} \) remains unchanged.
• Second term: Convert \( \frac{x}{2} \) to an equivalent fraction with denominator 4, resulting in \( \frac{2x}{4} \).

Once rewritten, the equation is \( \frac{x-5}{4}+\frac{2x}{4}=10 \). Now, you can combine the fractions: \( \frac{3x-5}{4}=10 \). This simplification facilitates solving the equation without dealing with disparate fractional terms.

Algebraic Solution

After arriving at the simplified equation \( \frac{3x-5}{4}=10 \), the next step is solving for \( x \) using algebraic methods. Here's how you do it:

First, eliminate the fraction by multiplying both sides of the equation by the denominator, in this case, 4. Doing so gives you \( 3x-5=40 \). This step clears the fraction and simplifies our equation to a basic linear form.

• Add 5 to both sides of the equation to isolate the \( 3x \) term: \( 3x=45 \).
• Divide both sides by 3 to solve for \( x \): \( x=15 \).

These operations help us systematically work our way to the value of \( x \), using basic algebraic principles like addition, subtraction, multiplication, and division. Solving the equation "algebraically" highlights a clear pathway to the solution without guesswork, ensuring precision and reliability.

Graphing Utility

Using a graphing utility is a fantastic way to verify algebraic solutions visually. In our example, after solving \( \frac{x-5}{4}+\frac{x}{2}-10=0 \) algebraically to find \( x=15 \), it's beneficial to see this result on a graph to confirm accuracy.

To do this, plot the function \( f(x)=\frac{x-5}{4}+\frac{x}{2}-10 \) on a graphing calculator or software. This graph should intersect the x-axis at the value we found — specifically, \( x=15 \).

• The point where the graph crosses the x-axis is the solution, showing where \( f(x)=0 \).
• You gain visual confirmation of the solution, reinforcing understanding and accuracy.

Graphing utilities bridge the gap between abstract algebraic solutions and tangible visuals, helping enhance comprehension especially for visual learners.