Problem 80 - Algebra A Combined Function

Question

For which of the following equations can we immediately use cross products to solve for \(x\) ? a. \(\frac{2-x}{5}=\frac{1+x}{3}\) b. \(\frac{2}{5}-x=\frac{1+x}{3}\)

Step-by-Step Solution

Answer

Equation (a) is a proportion where cross products can be used.

Step 1: Identify if each equation is a proportion

Proportions are equations that state two ratios are equivalent. We check if each equation is of the form \(\frac{A}{B} = \frac{C}{D}\).

Step 2: Analyze equation (a)

Equation (a) is \(\frac{2-x}{5} = \frac{1+x}{3}\). Both sides are fractions, thus this is a proportion and we can use cross products to solve for \(x\).

Step 3: Analyze equation (b)

Equation (b) is \(\frac{2}{5} - x = \frac{1+x}{3}\). The left side contains a subtraction from a fraction, making it not a direct proportion; therefore, we cannot immediately use cross products.

Key Concepts

Understanding Proportions

Proportions are a fundamental concept in mathematics that involve two ratios being equal to each other. You can think of a proportion as an equation with two fractions that are set equal. This relationship allows us to make sure that the two parts of the ratio change consistently. For example, if you know the ratio of baked goods made by a baker in a morning is equivalent to the ratio in the afternoon, you could find out how many croissants were baked if you knew the amount of cookies. Cross multiplication is a powerful tool here, as it lets us figure out unknown values when we know the others. As a key point: if you have an equation in the form \( \frac{A}{B} = \frac{C}{D} \), you can be assured that you’ve got a proportion that you can work with.

Decoding Equations

Equations are the backbone of algebra. An equation may look like just a set of numbers and letters, but it is a statement that two expressions are equal. Solving equations involves finding the value of unknowns (often represented as letters like \( x \) or \( y \)) that make the equation true. For instance, if you have \( x + 5 = 10 \), you could easily find that when \( x \) equals 5, both sides are equal. The equation settles the mystery of the unknown by dictating how different parts of the same statement relate to each other. When it comes to equations in proportions, knowing the formats is crucial. Equations where both sides are neat fractions allow the technique of cross multiplying to come into play and simplify the resolution process.

The Role of Ratios

Ratios provide a method to compare quantities. A ratio can show the relationship between two amounts, similar to how a proportion extends this into an equation. If you say you have a 2:3 ratio of apples to oranges, it tells you for every 2 apples, there are 3 oranges. The beauty of ratios is in their simplicity and wide range of applications. In real-world scenarios, ratios help in scaling recipes, converting currencies, or even explaining speeds. When these ratios translate into algebra through proportions, they transform the concept into a powerful tool for solving equations. Consider a proportion equation: \( \frac{2}{5} = \frac{x}{15} \), where you use ratios to figure out an unknown quantity.

Algebraic Techniques

Algebraic techniques involve using different mathematical operations and principles to manipulate equations to find an unknown value. These techniques are like the rules of a puzzle game, guiding you on how to move the pieces to reveal the answer. Cross multiplication is one such technique, especially handy for dealing with proportions. For a proportion like \( \frac{a}{b} = \frac{c}{d} \), cross multiply to get \( a \,*\, d = b \,*\, c \). This simplifies finding \( x \) in equations where you have fractions. But be cautious: cross multiplication is only applicable when you have a proper proportion, meaning each side of the equation is a corresponding fraction. If this is not observed, you might need additional steps to simplify your equation first.