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Chapter 2: Problem 36

Solve each equation, and check your solution. $$ 13 t+9=12 t+9 $$

Short Answer

Expert verified
The solution is t = 0.

Step by step solution

01

- Move terms involving t to one side

To isolate the variable t, subtract 12t from both sides of the equation: 13t + 9 - 12t = 12t + 9 - 12tThis simplifies to:t + 9 = 9
02

- Move constant terms to the other side

Next, subtract 9 from both sides of the equation to solve for t: t + 9 - 9 = 9 - 9 This simplifies to:t = 0
03

- Check the solution

Substitute t = 0 back into the original equation to verify the solution: 13(0) + 9 = 12(0) + 9 This simplifies to: 9 = 9 Since both sides of the equation are equal, t = 0 is indeed the correct solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variable Isolation
The first step in solving a linear equation is to isolate the variable. This means you want to get the variable (in this case, \( t \)) by itself on one side of the equation. In our example equation, \( 13t + 9 = 12t + 9 \), we start by moving all terms involving \( t \) to one side. We subtract \( 12t \) from both sides of the equation:

\[ 13t + 9 - 12t = 12t + 9 - 12t \]

This simplifies to:

\[ t + 9 = 9 \]

By moving like terms to one side, we make it easier to solve for the variable. Always perform the same operation on both sides of the equation to keep it balanced.
Linear Equations
A linear equation is an equation in which the highest power of the variable is one. The general form is:

\[ Ax + B = C \]

where \( A \), \( B \), and \( C \) are constants. In our example, both sides of the equation simplify to linear forms. After isolating the variable \( t \), we move constants to the other side to finally solve for \( t \). To accomplish this, we subtract 9 from both sides:

\[ t + 9 - 9 = 9 - 9 \]

This simplifies to:

\[ t = 0 \]

Linear equations are straightforward once you master the steps of isolating the variable and moving constant terms.
Solution Verification
Verifying your solution is a crucial step in solving linear equations. To ensure that \( t = 0 \) is indeed the correct solution, we substitute \( t = 0 \) back into the original equation:\( 13t + 9 = 12t + 9 \). Replacing \( t \) with 0, we get:

\[ 13(0) + 9 = 12(0) + 9 \]

This simplifies to:

\[ 9 = 9 \]

Since both sides of the equation are equal, \( t = 0 \) is indeed the correct solution. This step confirms that our process of variable isolation and manipulation of linear equations was done correctly.

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