Problem 11 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$\frac{3+5 x}{5}=\frac{4-x}{7}$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{-1}{40}\).

Step 1: Eliminate the Fractions

To eliminate the fractions, find a common denominator and multiply both sides of the equation by it. The denominators are 5 and 7, so the least common multiple is 35. Multiply both sides by 35:\[35 \cdot \left(\frac{3 + 5x}{5}\right) = 35 \cdot \left(\frac{4 - x}{7}\right)\]This simplifies to:\[7(3 + 5x) = 5(4 - x)\]

Step 2: Distribute on Both Sides

Distribute the numbers on both sides of the equation:\[7 \cdot 3 + 7 \cdot 5x = 5 \cdot 4 - 5 \cdot x\]Simplifying gives:\[21 + 35x = 20 - 5x\]

Step 3: Move Variable Terms to One Side

Add \(5x\) to both sides to move all terms containing \(x\) to one side:\[21 + 35x + 5x = 20 - 5x + 5x\]This simplifies to:\[21 + 40x = 20\]

Step 4: Isolate the Variable Term

Subtract 21 from both sides to isolate the term with \(x\):\[40x = 20 - 21\]This simplifies to:\[40x = -1\]

Step 5: Solve for x

Divide both sides by 40 to solve for \(x\):\[x = \frac{-1}{40}\]

Step 6: Verify the Solution

Substitute \(x = \frac{-1}{40}\) back into the original equation to verify. Plug back into:\[\frac{3 + 5\left(\frac{-1}{40}\right)}{5} = \frac{4 - \left(\frac{-1}{40}\right)}{7}\]Verify if both sides are equal:\[\frac{3 - \frac{5}{40}}{5} = \frac{4 + \frac{1}{40}}{7}\]Calculate both sides to ensure they are equal.

Key Concepts

Least Common Multiple

When you solve rational equations, it is crucial to first eliminate the fractions. One efficient way to do this is by using the Least Common Multiple (LCM) of the denominators. The LCM is the smallest number that both denominators can evenly divide into.

In our problem, the denominators are 5 and 7. The least common multiple of 5 and 7 is 35. By multiplying each term of the equation by 35, you effectively "clear out" the denominators.

• Multiply both sides of the equation by 35.
• This removes the denominators and allows for simpler equation solving.

This way, you transform a complicated fraction problem into something much more manageable.

Distributive Property

The distributive property is a key concept not only in solving rational equations, but in algebra in general. This property allows you to apply multiplication across terms inside parentheses.

In our example, after eliminating fractions, we have the equation: \[7(3 + 5x) = 5(4 - x)\]
To simplify:

• Multiply 7 by each term inside the parentheses: \(7 \cdot 3 + 7 \cdot 5x\)
• Similarly, multiply 5 by each term in the parentheses on the other side: \(5 \cdot 4 - 5 \cdot x\)
• The equation simplifies to \(21 + 35x = 20 - 5x\)

This step dramatically simplifies both sides of the equation and sets the stage for solving it.

Variable Isolation

Variable isolation is a technique that focuses on getting the variable, often represented by \(x\), alone on one side of the equation. This process involves moving terms strategically.

In our equation, the balance achieves this:

• Start by adding \(5x\) to both sides to gather all \(x\) terms on one side.
• You then have \(21 + 40x = 20\).
• Subtract 21 from both sides to further isolate \(x\) terms: \(40x = -1\)

With the variable isolated, you can now easily solve for \(x\). This method is a cornerstone for solving many types of equations.

Fractions in Equations

Fractions can complicate equations, but they are nothing to fear! They can be systematically managed and solved. In our exercise, fractions are present from the start.

• Fractions are eliminated by using the least common multiple, simplifying the equation.
• This technique converts a fraction equation into a linear equation without fractions, as demonstrated previously.
• By solving equations without fractions, we verify that the solution holds true when plugged back in.

Managing fractions effectively requires focus on obtaining a common denominator and correctly applying algebraic principles. This practice reinforces your equation-solving skills and boosts confidence when working with fractions in any algebraic context.