Problem 30 - Algebra and Trigonometry

Question

The given equation is either linear or equivalent to a linear equation. Solve the equation. \(\frac{2 x-1}{x+2}=\frac{4}{5}\)

Step-by-Step Solution

Answer

The solution is \( x = \frac{13}{6} \).

Step 1: Cross-Multiply the Fractions

Because we have a proportion, we can use cross-multiplication to eliminate the fractions. Multiply 2x - 1 by 5 and multiply 4 by x + 2:\[ 5(2x - 1) = 4(x + 2) \]

Step 2: Distribute the Multiplicands

Distribute the 5 and 4 to both terms inside their respective parentheses:\[ 10x - 5 = 4x + 8 \]

Step 3: Move Variables to One Side

Subtract 4x from both sides of the equation to get all x terms on one side:\[ 10x - 4x - 5 = 8 \]\[ 6x - 5 = 8 \]

Step 4: Isolate the x-Term

Add 5 to both sides to isolate the term with x:\[ 6x = 8 + 5 \]\[ 6x = 13 \]

Step 5: Solve for x

Divide both sides by 6 to solve for x:\[ x = \frac{13}{6} \]

Key Concepts

Cross-Multiplication

Cross-multiplication is a handy technique used to solve equations involving two fractions set equal to each other. When you have an equation in the form \(\frac{a}{b} = \frac{c}{d}\), you can eliminate the fractions by cross-multiplying. This means you will multiply the numerator of one fraction by the denominator of the other fraction, and set them equal to each other:

• Multiply \(a\) by \(d\) and\( b\) by \(c\).
• Set the two products equal: \(a \cdot d = b \cdot c\).

In our example, the equation \(\frac{2x - 1}{x + 2} = \frac{4}{5}\) uses cross-multiplication to remove the fractions:

• Multiply \(2x - 1\) by 5 and \(4\) by \(x + 2\).
• This gives us: \(5(2x - 1) = 4(x + 2)\).

Cross-multiplying allows you to transform the equation into a simpler form without fractions, making it easier to solve.

Distributive Property

The distributive property is a useful algebraic rule that helps simplify expressions involving parentheses. This property states that if you multiply a number by a sum, you can "distribute" the multiplication over each term inside the parentheses. It looks like this: \(a(b + c) = ab + ac\).In the context of solving the equation \(5(2x - 1) = 4(x + 2)\), we need to apply the distributive property to simplify it. Here’s how it works step by step:

• Apply the property to \(5(2x - 1)\) to get \(10x - 5\).
• Apply the property to \(4(x + 2)\) to get \(4x + 8\).

Once the distributive property is applied, our equation becomes \(10x - 5 = 4x + 8\). By distributing multiplication, expressions become more manageable and reveal the individual terms for comparison and simplification.

Isolating Variables

Isolating variables is a crucial aspect of solving equations in algebra. The goal is to rearrange the equation so that the variable, typically \(x\), is on one side of the equation by itself. This allows us to determine the variable's value easily.To isolate the variable in an equation, you generally follow these steps:

• Move all terms containing the variable on one side of the equation.
• Move all constant terms to the opposite side.
• Simplify both sides until the variable is by itself.

In our example, after distributing, we have \(10x - 5 = 4x + 8\). Here's how we isolate \(x\):

• Subtract \(4x\) from both sides: \(10x - 4x - 5 = 8\), simplifying to \(6x - 5 = 8\).
• Add 5 to both sides to remove the constant term: \(6x = 13\).
• Finally, divide both sides by 6 to solve for \(x\): \(x = \frac{13}{6}\).

By isolating the variable step-by-step, you can accurately solve linear equations and find the value of unknowns with ease.