Problem 71 - Elementary Algebra

Question

Solve. $$ 14=-38+10 x $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{26}{5} \) or \( x = 5.2 \).

Step 1: Add 38 to Both Sides

To isolate the term with the variable, add 38 to both sides of the equation to eliminate the constant term on the right side. This gives us: \( 14 + 38 = 10x \).

Step 2: Simplify the Left Side

Compute the sum on the left side: \( 14 + 38 = 52 \). Now, the equation is simplified to: \( 52 = 10x \).

Step 3: Divide Both Sides by 10

To solve for \( x \), divide both sides of the equation by 10, which is the coefficient of \( x \). This results in: \( x = \frac{52}{10} \).

Step 4: Simplify the Fraction

Simplify \( \frac{52}{10} \) by finding the greatest common divisor of 52 and 10, which is 2. Divide the numerator and the denominator by 2 to get: \( x = \frac{26}{5} \).

Step 5: Convert to Decimal Form

If required, convert the fraction \( \frac{26}{5} \) to decimal form by performing the division: \( x = 5.2 \).

Key Concepts

Algebraic Manipulation

In the world of solving linear equations, algebraic manipulation is a fundamental skill. It involves rearranging and simplifying expressions to achieve the desired form. Think of it as providing a clear path to find an answer. When you encounter an equation like \( 14 = -38 + 10x \), your goal is to change the equation step by step until the variable, in this case, \( x \), stands alone on one side.

During this process, you can add, subtract, multiply, or divide both sides of the equation by the same value. This keeps the equation balanced, like balancing a scale. For example, adding 38 to both sides is a classic move in algebraic manipulation to start isolating the variable. You'll find it surprisingly satisfying how these small tweaks can simplify what initially seems complicated.

Isolate Variable

Isolating the variable is where the magic happens in solving equations. It's all about getting our variable, \( x \), by itself on one side of the equation while maintaining equality between both sides. Initially, our variable was mixed with other numbers: \( 10x = 14 + 38 \).

The first step to achieve isolation is to remove constants from one side. Adding or subtracting a number that sits on the same side as the variable, like we did by adding 38 to both sides, helps to remove these unwanted terms. Eventually, when you also divide to separate the mix of coefficient and variable, as in dividing by 10, you perfectly isolate the variable. Isolating the variable brings clarity and makes the solution intuitively reachable.

Simplify Fractions

Simplifying fractions might seem like a small skill, but it boasts a profound impact when dealing with linear equations. In our solution, once \( x \) was isolated, we ended up with a fraction: \( \frac{52}{10} \). The job here is to reduce it to its simplest form, meaning it can't get any smaller without turning into a decimal or an integer.

To simplify, we find the greatest common divisor (GCD) for both the numerator and the denominator. For \( 52 \) and \( 10 \), the GCD is 2. Dividing both the top and bottom by this number gives you the clean fraction \( \frac{26}{5} \).

• First, identify the GCD.
• Divide both numerator and denominator by the GCD.
• Write down the simplified fraction.

This simple simplification purifies your solution, ensuring it's neat and precise.

Convert Fraction to Decimal

Converting fractions to decimals can make results more intuitive, especially for those who find decimals easier to interpret. Once you've simplified a fraction, like \( \frac{26}{5} \), it's straightforward to transform this into a decimal.

The process is simple: divide the numerator by the denominator. For \( \frac{26}{5} \), you'd perform the division 26 divided by 5, resulting in 5.2. This conversion helps present the answer in a different format, useful in different contexts where decimals are more relevant or easier to comprehend.

• Numerator ÷ Denominator = Decimal
• Apply long division if needed
• Write down the decimal equivalent

Converting fractions to decimals ensures flexibility in how you communicate mathematical solutions.