Problem 9 - Basic Technical Mathematics with Calculus

Question

Solve the given equations and check the results. $$1-\frac{t-5}{6}=\frac{3}{4}$$

Step-by-Step Solution

Answer

The solution is \( t = \frac{13}{2} \), and it is correct when checked against the original equation.

Step 1: Clear the Fraction

Start by eliminating the fraction on the left-hand side of the equation. Multiply each term of the equation by 6 to get rid of the denominator: \[ 6 \left( 1 - \frac{t-5}{6} \right) = 6 \cdot \frac{3}{4} \]This simplifies to: \[ 6 - (t-5) = \frac{18}{4} \]

Step 2: Simplify Both Sides

Simplify the fractions and distribute the terms:\[ 6 - t + 5 = \frac{18}{4} \]Combine like terms on the left side:\[ 11 - t = \frac{18}{4} \]Now simplify the fraction:\[ 11 - t = \frac{9}{2} \]

Step 3: Isolate the Variable

Isolate \( t \) by subtracting 11 from both sides:\[ -t = \frac{9}{2} - 11 \]Convert 11 into a fraction with the same denominator:\[ 11 = \frac{22}{2} \]\[ -t = \frac{9}{2} - \frac{22}{2} \]

Step 4: Solve for the Variable

Now perform the subtraction:\[ -t = \frac{9 - 22}{2} \]\[ -t = \frac{-13}{2} \]To solve for \( t \), multiply by -1:\[ t = \frac{13}{2} \]

Step 5: Check the Solution

Substitute \( t = \frac{13}{2} \) back into the original equation to ensure it's correct:\[ 1 - \frac{\left(\frac{13}{2}\right) - 5}{6} = \frac{3}{4} \]Simplify the expression inside the fraction:\[ \frac{\frac{13}{2} - 5}{6} = \frac{\frac{13}{2} - \frac{10}{2}}{6} = \frac{\frac{3}{2}}{6} = \frac{3}{2} \times \frac{1}{6} = \frac{3}{12} = \frac{1}{4} \]Substitute back into the equation:\[ 1 - \frac{1}{4} = \frac{3}{4} \]The left-hand side simplifies to \( \frac{3}{4} \), hence verifying the solution is correct.

Key Concepts

Understanding Algebra

Algebra is like a puzzle where numbers and letters meet! When you solve equations using algebra, you're finding the value of the unknown. In this equation, instead of a number, we have a letter, which is called a variable. Here, the variable is \( t \). The goal is to figure out what number \( t \) equals, making the entire equation true.
There are a few steps to solve algebraic equations:

• Identify terms: Look for numbers, variables, and signs in the equation.
• Rewrite the equation: Try to make it simpler or clearer.
• Use operations: Add, subtract, multiply, or divide to isolate the variable.

In our problem, algebra helps us find the value of \( t \) that makes both sides of the equation equal. This balance is what you solve for in any algebraic equation!
Let's dive deeper into how we handled the fractions next.

Fraction Simplification

Fractions can seem tricky at first, but they are just a way to describe parts of a whole. When you simplify fractions, you're making them easier to work with. In this exercise, we began by multiplying each term by 6 to clear out the denominator of the fraction on the left-hand side.
Here's why clearing the fraction works:

• By multiplying by 6, we eliminate the denominator, shifting the equation to \( 6 - (t - 5) = \frac{18}{4} \).
• The new equation is easier to deal with since it no longer has complex fractions involved.

Once you've simplified the fraction, you can better handle the numbers and focus on isolating the variable. Simplified fractions often reveal the 'true' size of the value in relation to the whole, making further calculations more straightforward. We'll next see how this simplification aids in isolating the variable.

Isolating the Variable

Isolating the variable means getting it by itself on one side of the equation. This process reveals the solution to your algebraic puzzle. In our exercise, we dealt with the equation \( 11 - t = \frac{9}{2} \). To solve for \( t \), we adjust the equation by performing operations that help us isolate \( t \).
Here's how it's done:

• First, subtract 11 from both sides, converting 11 to \( \frac{22}{2} \) to match denominators.
• This results in \( -t = \frac{9}{2} - \frac{22}{2} \).
• Perform the subtraction to get \( -t = \frac{-13}{2} \).
• Finally, multiply by -1, yielding \( t = \frac{13}{2} \).

Isolating the variable is essential because it gives you the exact value that solves the equation. This step is like unwrapping the final layer of a gift to discover what's inside, which in this case, is \( t \)! We confirm our result by checking the solution.

Checking the Solution

After solving the equation, it's important to verify that your answer is correct. This is called "checking the solution." For mathematical equations, especially involving variables and fractions, double-checking ensures accuracy.
Here's how we confirm the solution:

• Substitute back \( t = \frac{13}{2} \) into the original equation.
• Simplify the expression on the left-hand side to make sure it equals the right-hand side \( \frac{3}{4} \).
• Each calculation, if correct, will keep the balance, confirming \( \frac{3}{4} \) equals \( \frac{3}{4} \).

If both sides of the equation equal each other after substituting the solution, the answer is verified as correct. This process helps catch any errors and reinforces your understanding of algebraic processes. It’s like double-checking a locked door to make sure it's secure!