Problem 41 - Algebra A Combined Function

Question

Solve each equation for \(x .\) $$ \frac{x}{5}=\frac{x+2}{3} $$

Step-by-Step Solution

Answer

The solution is \(x = -5\).

Step 1: Cross-Multiply

To eliminate the fractions, we use cross-multiplication. Multiply both sides of the equation to get:\[x imes 3 = 5 imes (x + 2)\]This simplifies to:\[3x = 5(x + 2)\]

Step 2: Distribute and Simplify

Apply the distributive property on the right-hand side of the equation:\[3x = 5x + 10\]This expands to:\[3x = 5x + 10\]

Step 3: Move Variables to One Side

Subtract \(5x\) from both sides of the equation to get all \(x\) terms on one side:\[3x - 5x = 10\]This simplifies to:\[-2x = 10\]

Step 4: Solve for x

Divide both sides by \(-2\) to isolate \(x\):\[x = \frac{10}{-2}\]This simplifies to:\[x = -5\]

Step 5: Check the Solution

Substitute \(x = -5\) back into the original equation to ensure it satisfies the equation:\[\frac{-5}{5} = \frac{-5 + 2}{3}\]Calculate both sides:\[-1 = -1\]Since both sides are equal, \(x = -5\) is indeed the correct solution.

Key Concepts

Cross-Multiplication

Cross-multiplication is a technique to solve equations that involve fractions. It helps to clear the fractions by moving them across the equation in a visually balanced way. This method is especially useful when you have two fractions set equal to each other. Here's how you do it in practice:

• Take the numerator of the first fraction and multiply it by the denominator of the second fraction.
• Then, do the reverse: multiply the numerator of the second fraction by the denominator of the first.
• Set these two products equal to each other.

In our exercise, for the equation \( \frac{x}{5} = \frac{x+2}{3} \), we cross-multiply to get \( x \times 3 = 5 \times (x + 2) \). This clears the fractions and leads us to a simpler equation: \( 3x = 5(x + 2) \). Cross-multiplication effectively transforms the equation into one without fractions, making it simpler to solve.

Distributive Property

The distributive property is a fundamental math principle that helps in breaking down expressions. It states that multiplying a single term by a sum inside parentheses is the same as doing each multiplication separately, then adding. In formula terms, this looks like: \( a(b + c) = ab + ac \). This becomes useful when simplifying equations, such as in our current problem after cross-multiplying.We start with \( 3x = 5(x + 2) \). According to the distributive property, this expands to:

• Multiply \(5\) by \(x\), yielding \(5x\).
• Then multiply \(5\) by \(2\), resulting in \(10\).

So, applying the distributive property, \( 5(x + 2) \) becomes \( 5x + 10 \). This transformation is crucial as it allows the simplification of complex equations into more manageable forms.

Simplifying Equations

Simplifying equations involves reducing them to their simplest form, making it easier to identify the variable values. After using the distributive property, we got \( 3x = 5x + 10 \). To simplify this, the goal is to gather all variable terms on one side and constants on the other.Steps to simplify:

• Subtract \(5x\) from both sides to remove \(x\) from the right side.
• This yields \(3x - 5x = 10\), simplifying to \(-2x = 10\).

Now, the equation \(-2x = 10\) is straightforward to solve. By dividing both sides by \(-2\), we isolate \(x\) to find that \(x = -5\). Simplifying equations helps in revealing the true nature of what the solution is.

Checking Solutions

Checking the solution ensures that your calculated answer actually satisfies the original equation. It's an essential last step in solving equations to confirm the accuracy of your solution.Let's verify for our exercise:

• Substitute \(x = -5\) back into the original equation \( \frac{x}{5} = \frac{x+2}{3} \).
• This becomes \(\frac{-5}{5} = \frac{-5 + 2}{3}\), simplifying both sides gives \(-1 = -1\).

Since both sides of the equation are equal, it confirms that \(x = -5\) is indeed the correct solution. Checking solutions is not only a good practice for simple confirmation but also a tool for building confidence in your math skills.