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Chapter 4: Problem 70

Solve the equation. $$ 10-t=6 $$

Short Answer

Expert verified
The solution to the equation is \(t = 4\).

Step by step solution

01

Re-arrange the equation

To isolate t, take '10' to the other side of the equation to reverse its sign. So the equation \(10 - t = 6\) can be re-arranged as \(10 - 6 = t\). This is achieved by subtracting 6 from both sides.
02

Calculate the value of t

After re-arranging, subtract 6 from 10 which results in \(t = 4\). So the solution to the equation is \(t = 4\).

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Key Concepts

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

Algebraic Manipulation
To solve linear equations like the one given, mastering algebraic manipulation is key. Algebraic manipulation involves using operations like addition, subtraction, multiplication, and division to change the form of an equation without changing its equality. This process helps in simplifying equations:
  • Identify terms - Recognize which terms are constants, like numbers, and which involve variables, like \( t \).
  • Perform operations - Use the opposite operation to cancel out terms on both sides. For instance, if a term is subtracted from a variable, add it equally to both sides.
Using these strategies carefully allows the transformation of a complex equation into a simpler form. It's the foundation of any equation-solving process. Think of it as "cleaning up" the equation so that your variable stands out clearly.
Isolation of Variables
Isolation of variables is a critical step in solving equations. It refers to getting the unknown variable by itself on one side of the equation. In our example, the variable is \( t \), and it must be isolated to find its value. To do this, follow these steps:
  • Move unwanted terms - If a variable is being subtracted as in \( 10-t=6 \), you add or subtract from both sides to shift these unwanted terms. This "transfers" the terms to the other side.
  • Simplify step by step - Always move one term at a time to avoid confusion and errors. Here, isolating \( t \) allowed for finding its value simply.
Remember, the goal is to make sure the variable is alone on one side with all constants on the other. This clear "isolation" leads directly to determining the variable's value.
Equation Re-arrangement
Re-arranging an equation involves changing its structure to simplify solving it. With our equation, \( 10-t=6 \), the objective is to reconfigure it so that you can easily solve for \( t \). Here's how it's done:
  • Reverse operations - If a number is added on one side, subtract it from both sides. The same applies to subtraction, multiplication, and division.
  • Maintain balance - Always perform the exact same operation on both sides of the equation to preserve its equality. Moving terms without balance can lead to incorrect solutions.
This process is essential as it helps in managing and solving more complex equations efficiently. Once you get comfortable with re-arranging equations, solving them becomes a straightforward task.

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Most popular questions from this chapter

In Exercises 71-76, write an equation of the line that passes through the points. $$ (6,-1),(-3,-3) $$

In Exercises 21-28, sketch the graph of the linear inequality. $$ y \geq 2 x-1 $$

What point is often the most convenient test point to use when sketching the graph of a linear inequality?

In Exercises \(41-44\), determine the slope of the line. $$ y-2=5(x+3) $$

In Exercises \(5-8\), state whether the boundary of the graph of the inequality should be dashed or solid. $$ 2 x+3 y \leq 6 $$
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