Problem 29 - Algebra 1

Question

Solve the equation. $$288=16 t$$

Step-by-Step Solution

Answer

Therefore, the solution to the equation is \( t = 18 \).

Step 1: Identify the equation

The given equation is \(288=16 t\). We want to solve for \( t \).

Step 2: Isolate the variable

To isolate \( t \), divide both sides of the equation by 16. The equation then becomes \( t = 288 / 16 \).

Step 3: Compute the division

After carrying out the division, the equation simplifies to \( t = 18 \).

Key Concepts

Isolate the Variable

When solving linear equations, one of the most crucial steps is to 'isolate the variable'. This means manipulating the equation so that the variable you are solving for is on one side of the equation by itself. Consider the equation \(288=16t\). To isolate \(t\), we need to perform the same operation on both sides of the equation to keep it balanced. For multiplication or division, we do the inverse operation. Here, because \(t\) is multiplied by 16, we divide both sides by 16 to isolate it, resulting in \(t = \frac{288}{16}\). Remember, whatever you do to one side, you must also do to the other to maintain the equation's balance.

As an improvement to the exercise, it can be helpful to emphasize the importance of understanding the 'inverse operation' – this is key when it comes to isolating variables.

Equation Solving Step by Step

Approaching equation solving step by step helps break down the seemingly complex process into manageable parts. The given equation, \(288=16t\), illustrates a step-by-step approach. Firstly, identify the goal, which in this case is to find the value of \(t\). Secondly, use algebraic principles to isolate \(t\), as outlined in the previous section—dividing by 16. Lastly, compute the division to simplify the equation to \(t=18\). This sequential approach aids comprehension and reduces errors.

To enhance understanding, it might also be beneficial to include a narrative explaining why each step is necessary and to encourage students to write down each step as they perform it; this reinforces the learning process and serves as a useful checking mechanism.

Basic Algebra

At the heart of many mathematical problems is basic algebra, a branch of mathematics that deals with symbols and the rules for manipulating those symbols to solve equations. In basic algebra, understanding how to manipulate algebraic expressions according to the properties of arithmetic operations (addition, subtraction, multiplication, division) is fundamental. The problem \(288=16t\) is rooted in these principles. It reflects the distributive property, exemplified as division in this instance, and demonstrates the simplification of an algebraic expression to find a specific value for a variable.

Understanding these algebraic properties can greatly improve a student's ability to solve similar equations. Stressing the importance of foundational algebraic concepts, such as combining like terms and using the distributive property, can provide students with the tools they need to approach and solve a wide array of algebraic problems.