Problem 43 - Algebra 1: Concepts and Skills

Question

SOLVING WITH MENTAL MATH Use mental math to solve the equation. $$ 11 t=22 $$

Step-by-Step Solution

Answer

The solution to the equation \(11t = 22\) is \(t = 2\).

Step 1: Identify the Operation Needed

In the equation \(11t = 22\), \(t\) can be isolated by dividing both sides of the equation by 11.

Step 2: Apply the Operation to Both Sides

Applying the division operation to both sides of the equation: \(11t / 11 = 22 / 11\).

Step 3: Solve for the Unknown Variable

After performing the division, you get the unknown variable \(t\) on one side and the result of the operation on the other: \(t = 2\).

Key Concepts

Mental Math Strategies

To harness the power of mental math when solving equations, one must become adept at recognizing patterns and applying arithmetic operations in their head. Some key mental math strategies include breaking down complex problems into simpler ones, using number sense to estimate answers, and being familiar with times tables and the properties of operations. For example, in the equation \(11t = 22\), using mental math one can quickly recognize that 11 is a factor of 22. Thus, knowing that \(11 \times 2 = 22\), one can deduce that \(t\) must be 2 to make the equation true.

Another mental math strategy, particularly for multiplication and division, is to use known benchmarks. For instance, if you know that \(10 \times t\) would give a number close to 22, then you can infer that \(11 \times t\) gives you the exact answer. It's like fine-tuning your mental estimate to reach the correct solution. Equipped with these strategies, students can confidently tackle linear equations quickly and efficiently.

Solving Linear Equations

Solving linear equations is a foundational skill in algebra that involves finding the value of the unknown variable that makes the equation true. The process revolves around performing operations that will simplify the equation and isolate the variable. The typical steps include simplifying both sides of the equation, if necessary, and then using inverse operations to 'undo' any addition, subtraction, multiplication, or division that is affecting the variable.

In the equation \(11t = 22\), the goal is to get \(t\) by itself on one side of the equals sign. This is achieved by dividing both sides of the equation by 11. This operation is the inverse of multiplication and effectively balances the equation, leaving one with the solution \(t = 2\). Remember, what you do to one side of the equation, you must do to the other to maintain equality. Mastering this process is crucial for any further study in algebra and beyond.

Isolate the Variable

To solve for an unknown variable means to 'isolate' it on one side of an equation. Isolating the variable is the crux of solving simple linear equations and involves a few intuitive steps. The first step is to use inverse operations to undo any addition or subtraction affecting the variable. Next comes the undoing of multiplication or division. It's like peeling back layers to reveal the core value of the variable.

Using our given problem, \(11t = 22\), the variable \(t\) is initially tied up by being multiplied by 11. To free \(t\), divide both sides of the equation by 11, which is the inverse operation of multiplication. This leaves you with the isolated variable \(t = 2\). It's much like solving a puzzle, with each step bringing you closer to the 'picture' of the solution. Practicing this method creates a strong foundation for tackling more complex algebraic problems.