Problem 66 - Algebra 1: Concepts and Skills

Question

Use mental math to solve the equation. \(3 z=15\)

Step-by-Step Solution

Answer

The solution to the equation \(3 z = 15\) is \(z = 5\).

Step 1: Identify the equation

We are given the simple equation \(3 z = 15\). This equation tells us that 3 times a certain number \(z\) equals 15. Our task is to find the value of \(z\).

Step 2: Isolate the variable

To find the value of \(z\), we need to isolate it on one side of the equation. This can be achieved by dividing both sides of the equation by 3. So, the equation becomes \(z = 15/3\).

Step 3: Perform the calculation

By performing the division on the right-hand side, we find that \(z = 5\).

Key Concepts

Mental Math

Mental math allows us to solve problems in our head without needing paper or a calculator. In the equation provided, 3z = 15, you can quickly determine the solution mentally. Here’s how:

• Recognize that 3 multiplied by some number has to result in 15
• Use your knowledge of multiplication or division facts to find the number
• Consider simple multiples of 3, such as 3, 6, 9, 12, and finally 15

By listing these out, you can mentally identify that 3 multiplied by 5 equals 15. This quick process is the essence of using mental math effectively.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. In our given equation, algebra is at play when we identify "3z" as another form of algebraic expression.
In algebra, the equation 3z = 15 is a simple linear equation where:

• 3 is the coefficient - it scales the value of z.
• z is the variable that we are solving for.
• 15 is the constant - the result of the multiplication.

The goal of algebra is to isolate the variable, in this case, by performing the necessary operations, which guides our next steps.

Division in Equations

Division in equations is a crucial step to isolate the variable. In the scenario of 3z = 15:

• We have a multiplication of the variable we need to undo.
• To solve for z, divide both sides by the coefficient of z, which is 3.
• This results in z = 15/3.

Calculating 15 ÷ 3 gives us 5, showing that z = 5. This division step helps us find the unknown variable by reversing the multiplication originally performed, which is a foundational technique in solving equations.