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Chapter 2: Problem 239

Solve each linear equation. $$-(t-19)=28$$

Short Answer

Expert verified
t = -9

Step by step solution

01

- Distribute the negative sign

First, distribute the negative sign on the left side of the equation. This means changing the sign of each term inside the parentheses.o -(t - 19) = 28o -t + 19 = 28
02

- Isolate the variable term

Next, isolate the term with the variable by subtracting 19 from both sides of the equation.o -t + 19 - 19 = 28 - 19o -t = 9
03

- Solve for the variable

Now, solve for the variable by multiplying both sides by -1 to cancel out the negative sign.o -1(-t) = 9(-1)o t = -9

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Key Concepts

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

Understanding the Distributive Property
The Distributive Property is a fundamental concept in algebra. It states that multiplying a number by a sum is the same as multiplying the number by each addend and then adding the products.
In formula terms, this property is expressed as: \ a(b + c) = ab + ac \, where \(a\) is the number distributed to both \(b\) and \(c\).

Let's apply this to our exercise. We start with the equation:\ -(t - 19) = 28 \
This negative sign outside the parentheses needs to be distributed. It means we need to change the sign of each term inside the parentheses:
\ -t + 19 = 28 \
This step helps us simplify the equation by removing the parentheses and combining like terms.
Variable Isolation Techniques
The next step in solving a linear equation is to isolate the term containing the variable. Variable isolation means getting the variable by itself on one side of the equation.

Let's continue with our exercise where we have: \ -t + 19 = 28 \
To isolate \(-t\), we need to subtract 19 from both sides of the equation. This action keeps the equality balanced and moves the constant term to the right side:
\ -t + 19 - 19 = 28 - 19 \
Simplifying both sides, we get:
\ -t = 9 \
Now the term with the variable is isolated on one side of the equation.
Final Step: Solving for a Variable
The last step is solving for the variable. When solving for the variable, you perform operations to get the variable alone on one side of the equation.

In our exercise, we have: \ -t = 9 \
To solve for \(t\), we need to eliminate the negative sign. We do this by multiplying both sides by -1:
\ -1(-t) = 9(-1) \
This simplification gives us: \ t = -9 \
So, the solution to the original equation \ -(t - 19) = 28 \ is \ t = -9 \.
Through these steps, you can see how distributing, isolating the variable, and solving for the variable help us find the solution systematically.

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