Problem 16 - Algebra and Trigonometry

Question

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ 5 t-13=12-5 t $$

Step-by-Step Solution

Answer

The solution to the equation is \(t = 2.5\).

Step 1: Understand the Problem

We are given the equation \(5t - 13 = 12 - 5t\). Our goal is to find the value of \(t\). This involves isolating \(t\) on one side of the equation.

Step 2: Move Variable Terms to One Side

Add \(5t\) to both sides of the equation to move all terms involving \(t\) to one side: \[5t - 13 + 5t = 12 - 5t + 5t\] This simplifies to: \[10t - 13 = 12\]

Step 3: Isolate the Variable Term

Add 13 to both sides to isolate the variable term: \[10t - 13 + 13 = 12 + 13\] This simplifies to: \[10t = 25\]

Step 4: Solve for the Variable

Divide both sides by 10 to solve for \(t\): \[\frac{10t}{10} = \frac{25}{10}\] This gives us: \[t = 2.5\]

Step 5: Verify the Solution

Substitute \(t = 2.5\) back into the original equation to check: \[5(2.5) - 13 = 12 - 5(2.5)\] This simplifies to: \[12.5 - 13 = 12 - 12.5\] \[-0.5 = -0.5\], which confirms our solution is correct.

Key Concepts

Solving Linear Equations

Linear equations are fundamental in algebra and involve finding the unknown variable that makes the equation true. In our equation \(5t - 13 = 12 - 5t\), we aim to determine the value of \(t\). The equation is already linear because the variable \(t\) does not appear with exponents greater than one, and there is no multiplication between variables.

To solve linear equations, we follow certain steps consistently. The goal is to simplify each side of the equation and rearrange terms to make finding the variable straightforward. Here’s the general approach:

• Simplify each side separately if necessary by distributing and combining like terms.
• Move variable terms to one side of the equation, facilitating the isolation of the variable.
• Solve the simplified equation for the variable by performing arithmetic operations.
• Verify that the obtained solution satisfies the original equation.

These steps form a reliable method to tackle any solvable linear equation.

Variable Isolation

Variable isolation is a critical step in solving equations. We aim to isolate the variable, in this case, \(t\), on one side of the equation. By doing so, we make our job more straightforward because it becomes easier to see what operations are needed to determine the value of the variable.

In our exercise, after we have the equation \(10t - 13 = 12\), the next step is to isolate \(10t\) on one side. We achieve this by adding \(13\) to both sides, resulting in \(10t = 25\). Now, \(t\) is getting closer to being by itself. The final step is to divide both sides by \(10\) to solve for \(t\), giving \(t = 2.5\). Using operations like addition, subtraction, multiplication, and division strategically helps in these adjustments.

Remember: whatever operation you perform on one side of the equation, do the same to the other side. This keeps the equation balanced.

Equation Verification

After calculating the solution, it is crucial to verify if this solution is correct by substituting it back into the original equation. Verification ensures that no mistakes were made during calculations and confirms the solution's validity.

For our instance with \(t = 2.5\), substitute back into the initial equation:

• Original Equation: \(5t - 13 = 12 - 5t\)
• Substitute \(t = 2.5\): \(5(2.5) - 13 = 12 - 5(2.5)\)
• Calculate both sides: \(12.5 - 13\) results in \(-0.5\), and \(12 - 12.5\) also results in \(-0.5\).

The left and right sides equal, confirming \(t = 2.5\) is correct.

Verification not only reaffirms the correctness of our solution but also builds confidence in our problem-solving skills. By regularly practicing verification, you develop a habit of checking your work, which is an essential learning strategy.