Question

MENTAL MATH Write a question that could be used to solve the equation. Then use mental math to solve the equation. $$2 t-1=9$$

Short answer

The value of \(t\) that makes the equation \(2t - 1 = 9\) true is \(t = 5\).

Step-by-step solution

Formulate the question

The question that can be used to solve the equation \(2t - 1 = 9\) may be: 'What value of \(t\) makes the equation \(2t - 1 = 9\) true?'

Isolate the variable

To answer this question, achieve this by first adding 1 to both sides of the equation to cancel out the -1 on the left side. This results in the following equation: \(2t = 9 + 1\).

Solve for t

Solve the equation for \(t\) by dividing both sides of the equation by 2. This results in \(t = (9 + 1) / 2\).

Conduct mental math

Finally, apply mental math: \(t = 10 / 2\).

Key concepts

Mental Math

Mental math refers to the practice of performing arithmetic calculations in one's head, without the aid of a calculator or paper. It is a valuable skill that can save time and simplify many aspects of both academic and daily life. When applied to solving linear equations like the one provided, mental math can streamline the process of finding a solution.

For the given equation, \(2t-1=9\), one can quickly calculate that adding 1 to both sides will neutralize the -1, turning the equation into \(2t=10\). Further, it's a common mental math task to recognize that dividing 10 by 2 yields a result of 5. Therefore, the answer, \(t=5\), is obtained rapidly through mental computation.

Improving mental math capabilities often involves practice and familiarity with numerical patterns, such as recognizing multiples or common factors, which can significantly speed up the solving process for linear equations and other mathematical problems.

Isolate the Variable

The technique of isolating the variable is a fundamental concept in algebra, especially when solving linear equations. It revolves around performing operations to leave the variable we want to solve for by itself on one side of the equation, thus making the solution apparent.

In the context of the exercise, \(2t-1=9\), the goal is to isolate \(t\). The initial step is to eliminate the constant term on the same side as \(t\) by adding 1 to both sides, which balances the equation and simplifies it to \(2t=10\). Then, to fully isolate \(t\), both sides are divided by the coefficient of \(t\), which is 2. This leaves us with \(t=5\), thereby achieving the isolation of the variable.

This technique underlines the importance of inverse operations (addition/subtraction and multiplication/division) in the process of solving linear equations. Understanding this method enhances the ability to tackle more complex algebraic expressions effectively.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors and transformations, as well as the solving of linear equations and systems of linear equations. Linear equations describe a straight line in a two-dimensional space and frequently present relationships where one variable depends on another.

The given exercise, \(2t-1=9\), showcases a simple linear equation in one variable. In a broader context, linear algebra involves understanding concepts like matrices, determinants, and vector spaces, which extend beyond single equations to consider multiple variables and equations at once.

The process used to solve \(2t-1=9\) exemplifies the core methods of linear algebra—transformation and simplification. By performing algebraic operations to both sides of the equation, we transform it from its original state to a more manageable form, ultimately finding the value of \(t\) that satisfies the equation. Mastery of these fundamental principles prepares students for more advanced areas of mathematics and applications in physics, engineering, computer science, and economics.