Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-17,\) solve each equation or inequality. Each solution is an integer. $$ 12-3 y=18 $$

Step-by-Step Solution

Answer

The solution is \( y = -2 \).

Step 1: Isolate the term with the variable

Start with the equation: \[ 12 - 3y = 18 \]Our goal is to solve for \( y \). First, subtract 12 from both sides of the equation to isolate the term with \( y \) on one side:\[ 12 - 3y - 12 = 18 - 12 \]This simplifies to:\[ -3y = 6 \]

Step 2: Solve for the variable

Next, we need to solve for \( y \) by dividing both sides of the equation by -3:\[ -3y \, / \, (-3) = 6 \, / \, (-3) \]This gives:\[ y = -2 \]

Key Concepts

Understanding Linear Equations

A linear equation is an algebraic expression that represents a straight line when plotted on a graph. It usually has one or more variables, and the degree of the equation is one. This type of equation can be in the form of: \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable we need to find. Linear equations are fundamental because they help us understand the relationship between variables, significant when predicting outcomes or solving real life problems. In our exercise, the equation given is \( 12 - 3y = 18 \), which is a linear equation with one variable, \( y \).

• The equation involves subtraction and multiplication operations.
• Our objective is to find the integer solution for \( y \).

Linear equations can describe many situations, such as the cost of items, speed of an object, or even conversion of temperatures. Their flexibility and simplicity make them an essential part of algebra.

Isolation of Variable

Isolating the variable is a critical step in solving linear equations. It involves manipulating the equation to get the variable by itself on one side of the equation. This makes it easier to identify its value. In our example, we start with the linear equation \( 12 - 3y = 18 \). Our aim is to isolate \( y \), the variable, on one side of the equation. Here are the steps taken:

• Subtract 12 from both sides: This removes 12 from the left side, giving \( -3y = 6 \).
• The focus on symmetry helps in maintaining balance in the equation.

By isolating the variable, we bring forward clarity and eliminate extra complexities. This step helps facilitate solving as all terms involving \( y \) are brought together.

Solving Equations

Once the variable is isolated, we proceed to solve the equation to find its value. Understanding the nature of the operations involved is key here. After the isolation step, we have \( -3y = 6 \). The next task is to find \( y \)’s value:

• Divide both sides by \(-3\): This operation unwinds the multiplication involving \(-3\).
• Calculate \( y = 6 / (-3) \), simplifying to \( y = -2 \).

These calculations give us the integer solution of \( y = -2 \). Solving equations involves performing operations that balance the equation, such as addition, subtraction, multiplication, or division. With practice, this process becomes more intuitive, allowing you to tackle various types of algebraic problems confidently.