Problem 66 - Algebra 1: Concepts and Skills

Question

Use mental math to solve the equation. \(9-y=1\)

Step-by-Step Solution

Answer

The solution to the equation \(9-y=1\) is \(y=8\).

Step 1: Understand the Equation

In this equation \(9-y=1\), we want to solve for \(y\). That means we need to isolate \(y\) on one side of the equation.

Step 2: Isolate y

To isolate \(y\) on one side of the equation, we can subtract 1 from both sides. This gives us \(9-1=y\)

Step 3: Simplify

Our final step is to simplify the equation. When we subtract 1 from 9, we get 8. Therefore, our solution is \(y=8\).

Key Concepts

Mental Math

The practice of mental math allows students to perform basic arithmetic calculations without the need for a calculator or pen and paper. It emphasizes understanding and quick recall of mathematical concepts, which enhances problem-solving abilities.

Mental math is particularly useful for solving simple equations, such as the provided exercise, where you can visualize subtracting a number directly from another. To develop these skills, it is helpful for students to become familiar with:

• Addition, subtraction, multiplication, and division facts.
• Basic number sense and the ability to manipulate numbers easily.
• Various mental math strategies, like breaking numbers into more manageable parts.

Being proficient in mental math can greatly speed up the process of solving equations, especially during tests or situations where calculators are not available.

Isolate the Variable

When solving for a variable in an equation, the goal is to isolate the variable on one side to find its value. This involves using operations that 'undo' whatever is being done to the variable. If a variable is being subtracted, add the same number to both sides. If it's being multiplied, divide both sides by the same number, and so on.

For example, in the equation provided, we want to isolate the variable 'y'. The equation, \(9-y=1\), already implies that 'y' is being subtracted from 9, so we add 'y' to both sides and at the same time subtract 1 from the 9 to keep the balance of the equation, hence isolating 'y'.

Visualizing the Balance

Think of an equation as a balance scale. Whatever you do to one side, you must do to the other to keep the scale balanced. This balance is the core principle of algebra. By keeping the balance, you can correctly isolate the variable and solve the equation.

Simplify the Equation

To simplify the equation means to perform all possible basic operations to reduce the equation to its simplest form or to a point where the variable can be easily identified with its solution. It includes combining like terms, reducing fractions, and carrying out arithmetic operations.

In the case of the given problem, after isolating the variable by performing 9 - 1 to remove the constant from one side, we get \(8 = y\). This step simplifies the original equation and provides an easily identifiable solution. Simplifying an equation can also involve factoring expressions or getting rid of parentheses by using the distributive property.

Keep It Neat and Organized

While simplifying, it's important to write each step clearly. This helps in not only keeping track of what you have done but also makes it easier to identify any possible mistakes and understand the process when revisiting your work.