91Ó°ÊÓ

In solving linear equations, one of the primary goals is to isolate the variable, which means getting the variable by itself on one side of the equation. In the exercise we are looking at, the variable is \(b\). Initially, we have an equation: \(45b - 3 = 2\). To isolate \(b\), we need to "move" everything else away from \(b\).

This often involves rearranging terms by applying the opposite operation. Here, we begin by moving the constant term, \(-3\), to the other side of the equation. We do this by adding \(3\) to both sides because the opposite of subtraction is addition. This makes the transition:

• Original equation: \(45b - 3 = 2\)
• Adding \(3\): \(45b - 3 + 3 = 2 + 3\)
• Simplified to: \(45b = 5\)

By performing these operations, we have successfully moved \(-3\) to the other side, getting closer to having \(b\) isolated.

Mathematical operations, such as addition, subtraction, multiplication, and division, are the tools we use to manipulate equations and solve for variables. In linear equations, each operation has a corresponding opposite that helps to maintain balance.

Adding or subtracting the same number from both sides of the equation is crucial for isolating the variable without altering the equation’s equality. When we added \(3\) to both sides in our exercise, we

• ensured the equation remained balanced
• moved towards isolating the variable.

Another important operation is division. After isolating \(45b\) on one side, we need to remove the coefficient by dividing by it. This usage of division reverses the multiplication \(45b\) does with \(b\), letting us find the value of \(b\).

Therefore, knowing when and how to switch between operations is essential in rearranging and ultimately solving the equation.

Coefficients are the numbers placed before variables in equations. They indicate how many times that variable is multiplied. In our exercise, \(45\) is the coefficient of \(b\). Understanding coefficients is vital because they impact how we isolate the variable.

When we arrived at \(45b = 5\), the coefficient \(45\) hindered \(b\) from being isolated. The next logical step is to "get rid of" the coefficient by performing the opposite operation. Since \(45\) is multiplied by \(b\), we divide both sides of the equation by \(45\).

• Equation: \(45b = 5\)
• Dividing by \(45\): \(b = \frac{5}{45}\)
• Simplifying: \(b = \frac{1}{9}\)

By dividing, we effectively neutralized the coefficient to reveal the value of \(b\). Recognizing and handling coefficients correctly is key in managing and solving any algebraic equation.