Problem 36 - 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. $$\frac{x-3}{25}=\frac{x-5}{12}$$

Step-by-Step Solution

Answer

The solution to the equation \( \frac{x-3}{25} = \frac{x-5}{12} \) is \( x = 161/13 \). In the form \( f(x) = 0 \), it is written as \( \frac{x-3}{25} - \frac{x-5}{12} = 0 \). This can be verified using a graphing utility.

Step 1: Cross Multiply

Multiply both sides of the equation by 25*12 to eliminate the denominator of the fractions. This gives you: \(12*(x-3) = 25*(x-5) \)

Step 2: Distribute

Multiply the numbers in each parenthesis on both sides by the values outside the bracket. Simplify to get: \( 12x - 36 = 25x - 125 \)

Step 3: Isolate x

Get all the \( x \) terms on one side and the numbers on the other. Subtract \(12x\) from both sides and add 125 to both sides to isolate \( x \). This gives you: \( 25x - 12x = 125 + 36 \)

Step 4: Solve for x

Simplify both sides to get \( x \). This results in: \( x = 161/13 \)

Step 5: Write in form of f(x)=0

Finally, we write our equation in the format \( f(x) = 0 \). We do this by plugging in our value of \( x \) back into the original equation and subtracting one side from the other, like so: \( \frac{x-3}{25} - \frac{x-5}{12} = 0 \)

Step 6: Verify with graphing utility

Now you can use a graphing utility to plot both sides of the equation. The point where they intersect should match with our calculated \( x \) value.

Key Concepts

Cross Multiplication

Cross multiplication is a handy technique used to solve equations involving fractions. It involves multiplying each side of the equation by the other side's denominator. This process removes the fractions from the equation, allowing for a simpler expression to work with.
For the equation \( \frac{x-3}{25} = \frac{x-5}{12} \), cross multiplication is applied as follows:

• Multiply both sides by 25 to eliminate the denominator \(25\).
• Multiply both sides by 12 to eliminate the denominator \(12\).

The result of these operations is the equation \(12 \cdot (x-3) = 25 \cdot (x-5)\). This method is especially useful when dealing with proportions or multiple rational expressions. By simplifying the equation through cross multiplication, we streamline the process of finding the variable's value.

Isolation of Variables

Isolation of variables is a fundamental step in solving algebraic equations. The goal is to rearrange the equation so that one variable stands alone on one side of the equation. This often involves moving terms around and simplifying each side of the equation.
In the given problem, after applying cross multiplication, our new equation is \(12x - 36 = 25x - 125\). To isolate \(x\), follow these steps:

• Subtract \(12x\) from both sides: \(12x - 36 - 12x = 25x - 125 - 12x\), simplifying to \(-36 = 13x - 125\).
• Add 125 to both sides to deal with the constant terms: \(-36 + 125 = 13x - 125 + 125\).

This results in \(13x = 89\). Now, you can solve for \(x\) by dividing both sides by 13, yielding \(x = 89/13\). Isolation of variables is crucial because it allows us to pinpoint the exact value of the variable we are trying to solve.

Equation Verification with Graphing

Verifying solutions graphically provides a visual confirmation of your algebraic work. This method involves plotting the functions described by each side of the original equation. Once graphed, the point where the plots intersect represents the solution to the equation.
To verify the solution \(x = 89/13\) of the equation \(\frac{x-3}{25} = \frac{x-5}{12}\), first rewrite it as \(\frac{x-3}{25} - \frac{x-5}{12} = 0\). Using graphing software or a calculator, plot the function.
The graph should show a horizontal line at zero where the solution is plotted, specifically at \(x = 89/13\). This demonstrates that at this x-value, both sides of the equation balance each other, confirming the correctness of your solution.

Distributive Property

The distributive property is an algebraic principle used to simplify expressions. It states that for any numbers \(a\), \(b\), and \(c\), \(a(b+c) = ab + ac\). This property is essential when dealing with expressions involving brackets.
In our solved equation \(12(x-3) = 25(x-5)\), the distributive property was applied:

• For the left side: \(12(x - 3) = 12x - 36\).
• For the right side: \(25(x - 5) = 25x - 125\).

Using the distributive property helps break down more complex terms, thus making it simpler to isolate variables and solve the equation. This property proves incredibly useful because it reduces the complexity of the task by transforming products into sums or differences, which are easier to handle in algebraic manipulation.