Problem 33 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. $$ \frac{f}{-62}=103 $$

Step-by-Step Solution

Answer

Answer: f = -6386

Step 1: Multiply both sides of the equation by -62

Multiply both sides of the equation by -62 to get rid of the fraction: $$ -62 \cdot \frac{f}{-62}=103\cdot -62 $$ The -62 on the left side cancels out, leaving us with:

Step 2: Simplify the equation

Now that the fraction is gone, we have the equation: $$ f=-103 \cdot 62 $$ Calculate the value of -103 multiplied by 62:

Step 3: Calculate the result

Multiply -103 by 62: $$ f=-(103)(62)=-6386 $$

Step 4: Write the final answer

The solution to the equation is: $$ f=-6386 $$

Key Concepts

Solving Equations

Solving equations is like uncovering a mystery with math! It's all about finding out what number can replace the variable to make the equation true. An equation is a math sentence where two expressions are equal, and solving it means figuring out the value of the variable. To solve an equation, we usually perform operations to isolate the variable on one side. Common strategies include:

• Addition or subtraction to both sides
• Multiplication or division to both sides

In this exercise, we come across a conditional equation, which is an equation that may not be true for all values. We solved the equation \( \frac{f}{-62} = 103 \) by multiplying both sides by -62. This operation clears the fraction and helps us get to the value of \( f \). Don't forget: whatever you do to one side of the equation, you must do it to the other side too. This keeps the balance, just like weights on a scale.

Algebraic Fractions

Algebraic fractions are similar to the simple fractions we learned about in early math, but they contain variables. Understanding algebraic fractions can be tricky, but they're simply expressions where the numerator, the denominator, or both are algebraic expressions. Here's how to deal with them effectively in equations:

• To eliminate a fraction, multiply both sides of the equation by the denominator.
• Ensure that you simplify the equation after clearing the fraction.
• Be careful of division by zero – it's not allowed!

In our problem, the fraction \( \frac{f}{-62} \) initially contained the variable \( f \) on top. By multiplying both sides by -62, we successfully removed the fraction, making it easier to solve for \( f \). This process of removing fractions is essential in solving equations with algebraic fractions.

Multiplication of Integers

Multiplication of integers is key for solving many math problems. Integers are whole numbers, either positive or negative, including zero. Multiplying integers is straightforward, but careful attention is needed when signs are involved. Here’s how it works:

• If both integers are positive, the result is positive.
• If both integers are negative, the result is positive (the negatives cancel out).
• If one integer is negative and the other is positive, the result is negative.

In our step-by-step solution, we had to multiply \(-103\) by \(62\). Here, one number is negative, the other positive, so the result is negative: \(f = -6386\). Understanding how signs affect multiplication results is crucial in making sure your final answer is correct. It ensures you're on track in solving the full equation and achieving the right result.