Problem 7 - Algebra for College Students

Question

Solve each equation. \(4 y-3=21\)

Step-by-Step Solution

Answer

The solution to the equation is \(y = 6\).

Step 1: Isolate the Variable by Adding

To isolate the variable, we first need to eliminate the constant term on the left side. Add 3 to both sides of the equation:\[ 4y - 3 + 3 = 21 + 3 \]This simplifies to:\[ 4y = 24 \]

Step 2: Solve for the Variable by Dividing

Now that we have isolated the term with the variable, we need to solve for \(y\) by dividing both sides of the equation by 4:\[ \frac{4y}{4} = \frac{24}{4} \]This simplifies to:\[ y = 6 \]

Key Concepts

Solving Equations

When solving equations, we deal with expressions that involve an unknown quantity, often represented by a variable. Our aim is to find the value of this variable that makes the equation true. For example, consider the equation \(4y - 3 = 21\). Here, we want to determine what number \(y\) should be in order for the left side of the equation to equal the right side.

To achieve this, every step we take must maintain the balance of the equation. The principle here is whatever you do to one side of the equation, you must also do to the other. This might involve addition, subtraction, multiplication, or division. Understanding the process of solving equations helps us unlock countless other mathematical concepts and is foundational in learning algebra.

Whereas equations might look different, the steps to solve them are generally consistent, such as removing constant terms, and isolating and solving for the variable.

Isolate the Variable

Isolating the variable is one of the primary steps in solving linear equations. It involves reorganizing the equation so that the variable stands alone on one side. This makes it much easier to solve for its value.

In our example of \( 4y - 3 = 21 \), to isolate \( y \), we need to first eliminate the constant \(-3\) from the left side of the equation. To do this, we add 3 to both sides. This step ensures that the balance of the equation is maintained. After adding 3, the equation simplifies to \( 4y = 24 \).

By isolating \( y \), we turn our focus onto the term that contains \( y \), leading us to the final steps of solving the equation. Isolating the variable paves the way to unlocking its value efficiently.

Algebraic Manipulation

Algebraic manipulation is the process of relocating and rearranging terms within an equation to make it easier to solve. This technique is key in dealing with more complex equations as well.

In our equation \(4y = 24\), we manipulate the equation to find the value of \(y\). Since \(4y\) means 4 multiplied by \(y\), the next logical step is to divide both sides by 4. This division is an algebraic manipulation that simplifies \(4y\) to \(y\). The equation then becomes \(y = \frac{24}{4}\), which simplifies to \(y = 6\).

Algebraic manipulation allows us to work through equations even when they seem tough. It is a vital skill as it applies to every step of solving equations. Mastering manipulation techniques makes solving linear equations straightforward and more intuitive.