Problem 4 - Algebra Form and Function

Question

Solve \(g(x)=7\) given that \(g(x)=\frac{5 x}{2 x-3} .\)

Step-by-Step Solution

Answer

Answer: The value of x for which g(x) = 7 is x = 7/3.

Step 1: Set the function equal to 7 and solve for x

We need to find the value of \(x\) for which \(g(x) = 7\). So, we set the function equal to 7 and solve for \(x\): $$\frac{5x}{2x - 3} = 7$$

Step 2: Cross-multiply

Let's get rid of the fraction by cross-multiplying: $$5x = 7(2x - 3)$$

Step 3: Distribute 7 to the terms inside the parenthesis

Next, we distribute the 7 to both the terms inside the parenthesis: $$5x = 14x - 21$$

Step 4: Move the terms containing x to one side and the constant terms to the other side

To solve for x, we want to move all the terms containing x to one side and the constant terms to the other side of the equation: $$5x - 14x = -21$$

Step 5: Combine like terms and solve for x

Now, we can combine like terms and solve for x: $$-9x = -21 \Rightarrow x = \frac{21}{9}$$

Step 6: Simplify the fraction

To express the solution as a simplified fraction, we divide both the numerator and the denominator by their greatest common divisor (3): $$x = \frac{21}{9} = \frac{7}{3}$$ The solution to the equation \(g(x) = 7\) is \(x = \frac{7}{3}\).

Key Concepts

Solving Equations

Solving equations is a core part of algebra that involves finding the value of unknown variables. In our exercise, we are tasked with solving the equation \(g(x) = 7\) where \(g(x) = \frac{5x}{2x-3}\). This process requires setting up the equation in such a way that you can isolate \(x\). The first step involves writing the equation by setting the given function equal to 7:

• \(\frac{5x}{2x - 3} = 7\)

Next, aim to eliminate any fractions to simplify your work. Solving equations not only helps in algebra but also sharpens reasoning skills useful in real-life problem-solving. Remember, the goal is to have \(x\) by itself, ensuring it satisfies the original equation.

Cross-Multiplication

Cross-multiplication is a method used to eliminate fractions from equations. It involves multiplying each side of the equation by both the numerators and the denominators, so the equation no longer contains fractions. This technique is particularly useful when you have an equation involving a single fraction equal to a whole number or another fraction.In our example:

• After setting up the equation: \(\frac{5x}{2x - 3} = 7\), cross-multiplication helps by multiplying both sides by \(2x - 3\).
• This leads to: \(5x = 7(2x - 3)\).

Think of cross-multiplication as a tool to clear fractions quickly, simplifying the equation and making it easier to solve for the unknown variable. It is an essential skill in tackling rational equations effectively.

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form. After solving for \(x\), the obtained value could be a fraction that isn't in its simplest form. A simplified fraction is one where the numerator and the denominator have no common factors other than 1.In the solution:

• The fraction \(\frac{21}{9}\) emerges in the step before the final answer.
• Divide both the numerator and the denominator by their greatest common divisor (GCD), which is 3.
• This yields \(\frac{7}{3}\), which is the simplest form.

Simplifying fractions is important because it gives the most concise and clear expression of the solution. It's a fundamental skill ensuring that your final answer is both correct and elegantly expressed.