Problem 114 - Algebra A Combined Function

Question

Decide whether the given number is a solution of the given equation. \(\frac{x}{6}-3=5 ; 48\)

Step-by-Step Solution

Answer

Yes, 48 is a solution to the equation.

Step 1: Substitute the Given Number

Insert the number 48 into the equation for the variable \( x \). This gives us: \( \frac{48}{6} - 3 = 5 \).

Step 2: Simplify the Fraction

Calculate \( \frac{48}{6} \). This simplifies the equation to \( 8 - 3 = 5 \).

Step 3: Perform the Subtraction

Subtract 3 from 8, which results in \( 5 \).

Step 4: Compare the Result

The result we obtained is 5. Since the equation \( 5 = 5 \) is true, the number 48 satisfies the equation.

Key Concepts

Solving Equations

Solving equations is all about finding the value of the unknown variable that makes the equation true. In the exercise provided, the equation involves the variable \( x \). The task was to check whether the value provided (48) is a solution to this equation. When we say "solve an equation," we're looking for numbers that can replace the variable and make both sides equal. For example, in the equation \( \frac{x}{6}-3=5 \), replacing \( x \) with 48 should make both sides add up to the same value if it is a solution.Understanding the goal, which is to balance the equation, helps in correctly solving algebraic equations, as you essentially try to isolate the variable or verify if a given value works.

Substitution Method

The substitution method involves replacing a variable with a number to check if that number is indeed the solution. It's like testing if your guess is correct! In this problem, we substituted 48 for \( x \). This changes the equation from \( \frac{x}{6} - 3 = 5 \) to \( \frac{48}{6} - 3 = 5 \). The purpose here is to perform arithmetic operations to see if both sides of the equation match.Substitution is especially useful in verifying solutions or during algebraic manipulation when multiple unknowns are involved. It provides a straightforward path to check specific values by inserting them directly into the equation.

Fraction Simplification

Fraction simplification means reducing the fraction to its simplest form, which helps in making calculations easier. In our equation, the fraction \( \frac{48}{6} \) was simplified.The aim is to perform every operation such that the equation becomes simple and clear. For \( \frac{48}{6} \), you divide 48 by 6, which equals 8. This makes the equation much simpler: from \( \frac{48}{6} - 3 = 5 \) to \( 8 - 3 = 5 \).Understanding how to simplify fractions is crucial since it reduces complexity and makes the subsequent steps much more manageable.

Equation Verification

Equation verification is the final step where we check if the solution we obtained is correct. After performing all calculations, like substitution and simplification, we compare the results.Here, after simplification, we had \( 8 - 3 = 5 \), which held true because both sides equal 5. This confirmed that 48 is indeed a solution for the equation \( \frac{x}{6} - 3 = 5 \).Verification ensures that the solution fits the original equation, leaving no room for doubt. It's an essential part of problem-solving to affirm that the operations done prior were accurate and logical.