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Chapter 2: Problem 44

Solve each equation, and check your solution. $$ 1.9 t-6=0.9 t $$

Short Answer

Expert verified
t = 6

Step by step solution

01

- Move variables to one side

Subtract 0.9t from both sides of the equation to get all the terms involving t on one side. 1.9t - 0.9t - 6 = 0.9t - 0.9t 1.9t - 0.9t - 6 = 0
02

- Simplify the equation

Combine like terms on the left side of the equation. 1.9t - 0.9t is 1t or simply t. t - 6 = 0
03

- Solve for t

Add 6 to both sides of the equation to solve for t. t - 6 + 6 = 0 + 6 t = 6
04

- Check your solution

Substitute t = 6 back into the original equation to check if it satisfies the equation. 1.9(6) - 6 = 0.9(6) 11.4 - 6 = 5.4 5.4 = 5.4 Since both sides of the equation are equal, the solution t = 6 is correct.

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Key Concepts

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

Equation Simplification
In solving linear equations, the first step is often simplifying the given equation. This involves making the equation more manageable by combining like terms and removing any unnecessary terms.
For example, in our original exercise, we start with the equation: \(1.9t - 6 = 0.9t\).
To simplify it, we need to get all the terms involving the variable on the same side. So, we subtract \(0.9t\) from both sides:
**\(1.9t - 0.9t - 6 = 0\).**
By doing this, the equation becomes simpler and easier to solve. Simplification helps us focus on the important parts of the equation.
Variable Isolation
Variable isolation is the process of getting the variable (in this case, t) by itself on one side of the equation. This is crucial because it allows us to clearly see what the value of the variable is.
After simplifying the equation \(1.9t - 0.9t - 6 = 0\), we combine the terms involving \(t\):
**\(1.9t - 0.9t = 1t\)**.
This gives us:
**\(t - 6 = 0\)**.
Now, to isolate \(t\), we need to remove the -6 by performing the opposite operation. So, we add 6 to both sides of the equation:
**\(t - 6 + 6 = 0 + 6\)**, which simplifies to:
**\(t = 6\)**.
Now, the variable is isolated, and we can see that \(t\) equals 6.
Substitution
Substitution is the final step where we check our solution by plugging the value of the variable back into the original equation to see if it holds true.
From our solution, we have \(t = 6\). We substitute \(t\) back into the original equation:\(1.9t - 6 = 0.9t\):
**\(1.9(6) - 6 = 0.9(6)\)**.
This simplifies to:
**\(11.4 - 6 = 5.4\)**.
Since both sides of the equation are equal (\(5.4 = 5.4\)), our solution check confirms that \(t = 6\) is indeed correct.
This step is crucial to ensure accuracy, as it verifies that our solution works within the context of the given equation.

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