Problem 44 - Elementary Algebra Within Reach

Question

Solve the proportion. $$\frac{x-2}{4}=\frac{x+10}{10}$$

Step-by-Step Solution

Answer

The solution to the proportion is x = 10

Step 1: Cross-Multiplication

Cross-multiply to eliminate the fractions. This means to multiply the first numerator (top of fraction) by the second denominator and the first denominator by the second numerator. Here it will be \((x - 2) * 10 = (x + 10) * 4\).

Step 2: Simplification

Simplify the resulting equation by carrying out the multiplication. The result will be \(10x - 20 = 4x + 40\).

Step 3: Group the 'x' terms

Group the 'x' terms to one side of the equation and the constant terms to the other side. This results in \(10x - 4x = 40 + 20\).

Step 4: Solve for 'x'

Finally, simplify and solve for 'x'. This gives \(6x = 60\), and so 'x' is 10.

Key Concepts

Understanding Cross-Multiplication

Cross-multiplication is a powerful method used to solve proportion problems where a fraction equals another fraction. In proportions, cross-multiplication helps eliminate the denominators, making the equation easier to solve without fractions.
For example, in a proportion like \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction.
In our given problem, we have:

• First numerator: \( x-2 \)
• Second denominator: \( 10 \)
• First denominator: \( 4 \)
• Second numerator: \( x+10 \)

So, we cross-multiply to get:
\((x - 2) \cdot 10 = (x + 10) \cdot 4 \).
This results in an equation without fractions: \( 10(x - 2) = 4(x + 10) \). This method simplifies your math work by providing a clear path to an equation where you can solve for the unknown variable, \( x \).

Equation Simplification Techniques

Once you've used cross-multiplication to eliminate fractions, the resulting equation often benefits from further simplification. Simplification is the process of making an equation as straightforward as possible by performing operations like distributing and combining like terms.
In our equation from the cross-multiplication step, \( 10(x - 2) = 4(x + 10) \), the next step is to distribute the constants across the terms inside the parentheses.

• Distribute 10 across \( x-2 \) to get \( 10x - 20 \)
• Distribute 4 across \( x+10 \) to get \( 4x + 40 \)

This results in a simplified linear equation: \( 10x - 20 = 4x + 40 \).
The goal is to achieve a streamlined equation where you can easily see the next steps toward isolating and solving for the unknown variable. Remember always to perform identical operations on both sides of the equation to maintain balance.

The Art of Variable Isolation

Variable isolation is the technique used to solve for the unknown variable in an equation. It involves moving all terms involving the variable to one side of the equation and all constant terms to the other. This leaves the variable by itself, making it easy to identify its value.
From the simplified equation \( 10x - 20 = 4x + 40 \), follow these steps to isolate \( x \):

• Subtract \( 4x \) from both sides to bring all \( x \)-terms to one side: \( 10x - 4x = 40 + 20 \)
• Simplify to: \( 6x = 60 \)

Once the \( x \)-terms are on one side, you can isolate \( x \) completely by dividing both sides by the coefficient in front of \( x \):
\[ x = \frac{60}{6} \]
Which simplifies to \( x = 10 \).
Isolating the variable is a critical final step in solving algebraic equations, ensuring you reach an accurate solution.