Problem 18 - Fundamentals of Mathematics

Question

Solve each equation. $$ 6 y+3=-21 $$

Step-by-Step Solution

Answer

The solution to the equation is \( y = -4 \).

Step 1: Isolate the variable term

We want to solve for the variable, which is \(y\) in the equation \(6y + 3 = -21\). Start by getting rid of the constant term on the left side of the equation by subtracting 3 from both sides: \[ 6y + 3 - 3 = -21 - 3 \]The equation simplifies to:\[ 6y = -24 \]

Step 2: Solve for the variable

Now that the equation is simplified to \(6y = -24\), divide both sides by 6 to solve for \(y\):\[ \frac{6y}{6} = \frac{-24}{6} \]This yields:\[ y = -4 \]

Key Concepts

Isolating the Variable

In the world of algebra, solving equations often starts with a crucial step: isolating the variable. The aim is to have the variable on one side of the equation, usually the left, and everything else on the other side. This process makes it easier to see what the variable is equal to.

In our example equation, \(6y + 3 = -21\), the variable we are trying to isolate is \(y\). When isolating the variable:

• First, identify the variable term and any constant terms associated with it.
• Use arithmetic operations, such as addition or subtraction, to "move" constant terms to the other side of the equation.
• Ensure you perform the same operation on both sides to maintain balance in the equation.

By isolating the variable \(y\), we simplify the equation to better see what \(y\) is equal to.

Constant Term

A constant term is a fixed number in an equation, like the 3 in \(6y + 3 = -21\). It does not change and is not multiplied by the variable. When solving equations, one goal is to get rid of this constant term from the same side as the variable term.

In our example:

• We have the constant 3 right next to the variable term \(6y\).
• To eliminate this constant, we subtract 3 from both sides of the equation.
• This action helps to free the variable term from any additions or subtractions.

Handling constants effectively helps simplify the equation, preparing us to locate the value of the variable with ease.

Simplification

Simplification means making the equation easier to handle. It often involves combining like terms or eliminating unnecessary numbers. With fewer terms, the equation becomes a straightforward expression to solve. Simplifying an equation is like clearing clutter from a messy room; it becomes easier to find what you are looking for.

In our exercise, we simplify the equation by:

• Removing the constant term through subtraction, \(6y + 3 - 3 = -21 - 3\), resulting in \(6y = -24\).
• This gives us a simpler form of the equation, focusing only on the variable part.
• It allows us to see that the only step left involves dealing directly with the variable term, \(6y\).

Simplification is essential in making the problem more approachable, enabling us to identify the next steps toward finding the solution.

Division Operation

The division operation is perhaps one of the most critical steps in isolating a single instance of the variable, especially when the variable is multiplied by a number. In the simplified equation, \(6y = -24\), \(y\) is multiplied by 6. To solve for \(y\), we need to "undo" this multiplication by using division.

In our example:

• We perform the division operation across the entire equation by dividing both sides by the number 6.
• Mathematically, this looks like: \(\frac{6y}{6} = \frac{-24}{6}\).
• Performing the division yields \(y = -4\).

Division reduces the equation to its simplest terms, isolating the variable as a standalone number. It's like having all pieces of a puzzle removed, leaving only the solution itself.