Problem 40 - Algebra 1

Question

Write a question that can be used to solve the equation. Then use mental math to solve the equation. \(a-5=19\)

Step-by-Step Solution

Answer

The value of 'a' is 24.

Step 1: Understand the equation

The given equation is \(a-5=19\). The goal is to determine the unknown 'a'. This will be done by isolating 'a' on one side of the equation.

Step 2: Rearrange the equation to isolate 'a'

In order to isolate 'a' on one side of the equation, you can add 5 to both sides of the equation to maintain the equilibrium. This will result in the equation, \(a-5+5=19+5\), which simplifies to \(a=24\).

Key Concepts

Isolating Variables

Isolating variables is the first important step in solving linear equations. The idea is to have the unknown variable on one side of the equation and all other numbers on the opposite side. This process helps us clearly see what value the variable holds.
To achieve isolation, we perform the same operation on both sides of the equation to maintain balance or equivalence. In our exercise, the equation is \(a - 5 = 19\). To isolate \(a\), we need to undo the subtraction of 5.

• We add 5 to both sides of the equation, transforming it into \(a - 5 + 5 = 19 + 5\).
• This simplifies down to \(a = 24\), effectively isolating \(a\).

When you isolate the variable, remember that every action on one side of the equation must be mirrored on the other side. This fundamental concept helps maintain the truth of the equation throughout the process.

Mental Math

Mental math plays a significant role in solving equations quickly and effortlessly. It involves solving mathematical problems in your head without the aid of calculators or writing calculations down. In the case of simple linear equations, like \(a - 5 = 19\), mental math can make problem-solving fast.
Consider these mental steps when tackling such an equation:

• If subtracting a number gives you a result, add the same number to find the original one. Here, by adding 5 to the result of 19, you mentally conclude that \(a\) must be 24.
• This not only speeds up the process but also enhances one's ability to manipulate numbers mentally.

Practicing mental math can develop your numeracy skills, enabling you to handle mathematical concepts more fluidly and intuitively.

Equation Rearrangement

Equation rearrangement is a useful skill that allows us to modify equations to make them easier to solve. This often involves moving terms around to set them in a format that readily yields the value of the unknown.
The rearrangement can be simplified into a series of logical steps:

• Identify which term needs to be isolated (in our example, the variable \(a\)).
• Determine the operations needed to isolate the term (here, add 5 to both sides).
• Perform the operations, whittling down the equation step-by-step until the target variable is by itself on one side of the equation.

In summary, rearranging equations is about mastering how terms interact with each other. Understanding this concept leads to a better grasp of how to systematically find solutions to problems.