Problem 20 - Algebra and Trigonometry

Question

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 is \(t = 2.5\).

Step 1: Understand the Equation

The given equation is \(5t - 13 = 12 - 5t\). It is necessary to solve for \(t\), and since it is linear, solving it involves isolating \(t\).

Step 2: Move Terms Involving t to One Side

Add \(5t\) to both sides of the equation to bring all \(t\) terms on one side:\[5t - 13 + 5t = 12 - 5t + 5t\]Simplifying gives:\[10t - 13 = 12\]

Step 3: Isolate the Variable Term

Add 13 to both sides to isolate the \(10t\) term:\[10t - 13 + 13 = 12 + 13\]Simplifying gives:\[10t = 25\]

Step 4: Solve for t

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

Step 5: Verify the Solution

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

Key Concepts

Solving Linear Equations

Linear equations are equations that can be written in the form \(ax + b = c\) where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable we want to solve for. Solving a linear equation involves finding the value of the variable that makes the equation true. It's all about making the variable, in this case \(t\), on one side of the equation by itself.

To solve the example equation \(5t - 13 = 12 - 5t\), the goal is to get all terms involving \(t\) on one side and constants on the other. This generally involves:

• Adding or subtracting terms to both sides to move the variable terms to one side.
• Combining like terms to simplify.
• Using operations like addition, subtraction, multiplication or division to isolate the variable.

This method ensures a step-by-step approach to keep the equation balanced while working towards the solution.

Equation Verification

Once you have found a solution for the variable in a linear equation, it's crucial to verify it to ensure the solution is accurate. Verification involves substituting the solution back into the original equation to check if both sides are equal. This step helps confirm the correctness of the solution through a practical approach.

In our example, substituting \(t = 2.5\) back into the original equation \(5t - 13 = 12 - 5t\) checks if the left side equals the right side:

• Substitute: \(5(2.5) - 13 = 12 - 5(2.5)\).
• Calculate both sides: \(12.5 - 13 = 12 - 12.5\).
• The equality holds since \(-0.5 = -0.5\).

Great! This shows that the value \(t = 2.5\) is indeed the correct solution.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to isolate the variable. This is often necessary when dealing with linear equations. It involves using a series of logical and mathematical steps to modify the equation while still keeping it balanced.

Let's look at how algebraic manipulation was used in our example:

• Adding \(5t\) to both sides: This moved all \(t\) terms to one side, resulting in a simpler equation \(10t - 13 = 12\).
• Adding 13 to both sides: Isolating the \(10t\) term made it more straightforward to solve for \(t\). The equation now becomes \(10t = 25\).
• Dividing by 10: Finally, dividing both sides by 10 truly isolates \(t\), resulting in \(t = 2.5\).

Choosing these specific operations helps in systematically reducing the equation to its simplest form, thereby finding the solution efficiently.