Problem 17 - Algebra 2 and Trigonometry

Question

Solve each proportion for the variable. \(\frac{4 x-8}{3}=\frac{8}{x-3}\)

Step-by-Step Solution

Answer

The solution is \( x = 5 \).

Step 1: Identify the Proportion

The given exercise is a proportion - two fractions set equal to each other. The problem is \( \frac{4x-8}{3} = \frac{8}{x-3} \). We need to solve this proportion for the variable \( x \).

Step 2: Cross-Multiply the Fractions

To solve the proportion, start by cross-multiplying the fractions. This means multiply the numerator of the first fraction by the denominator of the second fraction and vice versa. This gives us \( (4x - 8)(x - 3) = 8 \times 3 \).

Step 3: Simplify and Multiply

First, compute the multiplication on the right side of the equation: \[ 8 \times 3 = 24 \]Next, apply the distributive property to the left side:\[ (4x - 8)(x - 3) = 4x(x) - 4x(3) - 8(x) + 8(3) \]This simplifies to:\(4x^2 - 12x - 8x + 24\).

Step 4: Combine Like Terms

Now, combine like terms in the equation:\[ 4x^2 - 12x - 8x + 24 = 24 \]This simplifies to:\[ 4x^2 - 20x + 24 = 24 \].

Step 5: Set Up and Simplify the Quadratic Equation

Subtract 24 from both sides to form the standard quadratic equation:\[ 4x^2 - 20x + 24 - 24 = 0 \]This simplifies to:\[ 4x^2 - 20x = 0 \].

Step 6: Factor the Quadratic Equation

Factor out the common term from the quadratic equation:\[ 4x(x - 5) = 0 \].

Step 7: Solve for the Roots

Set each factor equal to zero:1. \( 4x = 0 \) gives \( x = 0 \).2. \( x - 5 = 0 \) gives \( x = 5 \).

Step 8: Verify the Solutions

To ensure both solutions are valid, substitute back into the original proportion. For \( x = 0 \), the denominator becomes negative or undefined, so it is invalid in the context of the original equation. Verify for \( x = 5 \): - Substitute \( x = 5 \) into the original equation: - Left side: \( \frac{4(5) - 8}{3} = \frac{12}{3} = 4 \) - Right side: \( \frac{8}{5-3} = \frac{8}{2} = 4 \) Both sides equal, confirming \( x = 5 \) is a valid solution.

Key Concepts

Cross-Multiplication

Cross-multiplication is a wonderful tool for solving proportions. When you have two fractions set equal to one another, like \( \frac{4x-8}{3} = \frac{8}{x-3} \), you can use cross-multiplication. The key idea is to multiply the diagonal terms. Here's how it works:

• Multiply the numerator of the first fraction by the denominator of the second: \((4x - 8)(x - 3)\).
• Multiply the numerator of the second fraction by the denominator of the first: \(8 \times 3\).

After cross-multiplying, you'll have an equation without fractions, which is much easier to handle. Finish up by solving for the variable as needed. This technique is particularly helpful when you run into proportions in algebra and it simplifies your work tremendously.
Just remember, cross-multiplying helps balance the scales, turning a fraction-filled equation into something more approachable.

Quadratic Equation

A quadratic equation is any equation that can be rearranged into the standard form \(ax^2 + bx + c = 0\). In our scenario, after applying the distributive property and simplifying, we obtained \(4x^2 - 20x = 0\). This qualifies as a quadratic equation because it features a term with \(x^2\), a linear term \(x\), and a constant term, which is zero in this simplified case.

• To solve a quadratic equation, look for methods such as factoring, the quadratic formula, or completing the square.
• Quadratic equations often have two solutions, corresponding to the points where the graph intersects the x-axis.

In this example, we solve the quadratic by factoring, which is sometimes the easiest approach. Understanding quadratic equations opens doors to solving many real-world problems, as they frequently describe natural phenomena, like projectile motion or areas.

Distributive Property

The distributive property is a nifty tool that helps simplify expressions that involve multiplying a term by a binomial, like \((4x - 8)(x - 3)\). It's written as \(a(b + c) = ab + ac\). Here's what we did:

• Multiply each term inside the parenthesis by \(4x\): \(4x \cdot x\) and \(4x \cdot -3\).
• Multiply each term by \(-8\): \(-8 \cdot x\) and \(-8 \cdot -3\).

This transforms the expression into \(4x^2 - 12x - 8x + 24\). Utilizing the distributive property allows you to handle complex expressions by breaking them into simpler parts. It simplifies your work process and ensures that you retain all components of the original expression. Mastering this property allows you to tackle algebra with confidence.

Factoring Quadratics

Factoring quadratics is the process of breaking down a quadratic equation into a product of simpler terms. In our case, \(4x^2 - 20x = 0\) was factored into \(4x(x - 5) = 0\). Factoring tells us the values of \(x\) that make the equation equal to zero.Here's a step-by-step discussion:

• First, identify a common factor from all the terms and factor it out, here it was \(4x\).
• Then, see what's left inside the parenthesis and factor further if possible, yielding \(x - 5\).

Set each factor to zero and solve for \(x\), giving us the solutions. Factoring is an essential method for solving quadratic equations without delving into more complex formulas. It's particularly useful in both algebra and calculus, especially when graphing polynomial functions. By mastering factoring, you gain insight into the behavior of quadratic functions and how they interact with various axes or points on the graph.