Problem 79 - Algebra 1

Question

Solve the equation. $$ 9-y=4 $$

Step-by-Step Solution

Answer

The solution to the equation is \(y = 5\).

Step 1: Understand the problem

The equation given is \(9 - y = 4\). The aim is to solve for 'y', which means isolating 'y'.

Step 2: Transpose

Transpose '-y' to the right side of the equation and '4' to the left side. In this process, change the signs. So it will become \(9 - 4 = y\).

Step 3: Simplify

Simplify \(9 - 4\) to find 'y'. So, \(y = 5\).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. In algebra, letters like \(x\) and \(y\) represent numbers. The goal is often to find the value of these letters—referred to as variables.

• Understanding algebra involves recognizing patterns and using methods to solve equations.
• Equations are mathematical statements that show the equality between two expressions.
• In our exercise, we use the equation \(9 - y = 4\). This means the expression "9 minus y" is equal to 4.

Algebra can be seen as a powerful tool that provides us with a language to model real-world situations and solve problems efficiently.

Isolation of Variables

Isolation of variables is a fundamental process in algebra, especially when solving equations. It involves manipulation of the equation to get the variable of interest by itself on one side of the equation. Here's how it works:

• To isolate a variable, you usually perform the same operation on both sides of the equation. This keeps the equation balanced.
• In our example \(9 - y = 4\), we need to isolate \(y\). To do this, we shifted terms across the equation. This involves changing the signs when moving a term from one side to the other.
• We transposed \(-y\) to the right side and \(4\) to the left, resulting in \(9 - 4 = y\).

By following these steps, the variable \(y\) becomes isolated, making it straightforward to find its value.

Linear Equations

Linear equations are equations between two variables that produce a straight line when graphed. They are characterized by having variables raised only to the power of one. Here's what you need to know:

• They often appear in the form \(ax + b = c\).
• In the exercise, the equation \(9 - y = 4\) is a simple linear equation with two terms on each side.
• The primary goal is to solve for the variable, which involves isolating it as discussed earlier.
• Linear equations are straightforward because they have only one solution in this context. Thus, once we find that \(y = 5\), we know it's the solution.

These equations help in simplifying and solving everyday mathematical problems due to their predictable nature.