Problem 39 - Algebra 1: Concepts and Skills

Question

SOLVING WITH MENTAL MATH Use mental math to solve the equation. $$ 4 p=36 $$

Step-by-Step Solution

Answer

The solution to the equation is \(p = 9\).

Step 1: Identify the equation

From the given exercise, the equation to solve is \(4p = 36\). This can be interpreted as 4 times some number \(p\) is equal to 36.

Step 2: Use reverse operations to solve

To isolate \(p\) on one side of the equation, do the opposite or reverse of multiplication which is division. Divide both sides of the equation by 4.

Step 3: Solve for p

When you divide both sides by 4, the equation becomes \(p = 36 ÷ 4\). Solving this gives \(p = 9\).

Key Concepts

Mental Math Techniques

Mental math isn't just about quick calculations; it's also a foundational skill crucial for efficiently solving equations. To master this skill, it's essential to visualize simple arithmetic operations and number relationships. For example, when you encounter multiplication, like in the equation from the exercise (4p = 36), you can quickly recall multiplication tables to find the answer. Understanding factors and multiples can also aid in solving such problems mentally.

Here are some key mental math strategies:

• Break down numbers into smaller, more manageable sums or factors.
• Use known multiplication facts and patterns (e.g., anything multiplied by 4 is double what it would be if it were multiplied by 2).
• Employ estimation to check the reasonableness of your answer.
• Apply the distributive property to simplify complex multiplication.

With these techniques, you can transform seemingly complex problems into simpler math that you're more comfortable solving in your head.

Isolate Variable in Equation

When solving equations, the core objective is to isolate the variable, to find its value. This process involves performing operations that will simplify the equation and get the variable by itself on one side. In the equation from our example, 4p = 36, we're looking to find the value of 'p.'

To isolate 'p,' you should do the following:

• Determine what operation is currently being applied to the variable (in this case, multiplication).
• Apply the inverse operation to both sides of the equation (since 'p' is being multiplied by 4, you'll divide both sides by 4).
• Ensure everything you do to one side of the equation is also done to the other side to maintain balance.
• Once isolated, 'p' should be by itself on one side, giving you 'p = 9' as the solution in our exercise.

By isolating the variable, you've effectively simplified the equation to its most basic form, making the solution clear.

Reverse Operations in Algebra

Reverse operations in algebra are essentially the 'undoing' of what has been done to a variable. In our exercise, we have 4p = 36, and to solve for 'p,' we need to reverse the multiplication of 'p' by 4. This is where the concept of inverse operations comes into play – multiplication's opposite is division, addition's opposite is subtraction, and so forth.

Here's how reverse operations work in practice:

• To undo multiplication, divide by the same number.
• To undo division, multiply by the same number.
• To undo addition, subtract the same number.
• To undo subtraction, add the same number.

By applying the inverse operation, we maintain the equation's balance and isolate the variable to find its value. Consistent practice with these reverse operations builds a solid understanding of algebraic principles and improves problem-solving skills.