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Chapter 2: Problem 69

\(5 t+3+2 t-6 t=4+12\)

Short Answer

Expert verified
t = 13

Step by step solution

01

Combine Like Terms

First, combine the like terms on the left side of the equation. Combine the terms involving the variable t (5t, 2t, -6t) and the constant terms (3): 5t + 2t - 6t + 3 = 4 + 12
02

Simplify the Equation

Next, simplify the combined terms: 5t + 2t - 6t + 3 becomes (5 + 2 - 6)t + 3 which simplifies to: t + 3 = 16
03

Isolate the Variable

Now, isolate the variable t by subtracting 3 from both sides of the equation: t + 3 - 3 = 16 - 3 This simplifies to: t = 13

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Key Concepts

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

Combining Like Terms
When solving linear equations, combining like terms is crucial. Like terms are terms in an equation that have the same variable raised to the same power. For instance, in the equation: 5t + 3 + 2t - 6t = 4 + 12The terms 5t, 2t, and -6t are like terms because they all contain the variable t. By grouping these together, we can simplify the equation. Here's how:
  • First, arrange and combine the t-terms: 5t + 2t - 6t.
  • Next, handle any constants separately. Here, we have the constant term +3.
Putting these together results in:5t + 2t - 6t + 3 = 4 + 12This step simplifies future calculations and makes the subsequent steps more straightforward.
Isolating Variables
After simplifying like terms, the next step is to isolate the variable. Isolating a variable means getting the variable alone on one side of the equation. Let's continue with our example:From the previous step, we have the simplified equation:t + 3 = 16To isolate t, we must remove the +3 from the left side. We can do this by performing the same operation on both sides of the equation. This ensures the equation remains balanced. So, we:
  • Subtract 3 from both sides: t + 3 - 3 = 16 - 3
  • This simplifies to: t = 13
Now, the variable t is isolated, and we have our solution: t = 13. Isolating variables helps in easily identifying the solution to the equation.
Simplification Steps
Simplification is all about reducing the equation to a more manageable form. This involves combining like terms, performing arithmetic operations, and isolating variables as shown before. The goal is to transform a complex equation into a simpler one that is easier to solve.Starting from our initial equation:5t + 3 + 2t - 6t = 4 + 12, let's break down the simplification process:
  • Combine like terms: (5t + 2t - 6t) + 3 = 4 + 12
  • Simplify inside the parentheses: (5 + 2 - 6)t + 3 = 16
  • This reduces to: t + 3 = 16
  • Finally, isolate the variable by subtracting 3 from both sides: t = 13
Each step of simplification reduces the complexity of the problem, making the path to the solution clearer and more straightforward.

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