Problem 69 - Elementary Algebra

Question

Solve. $$ -3=3.36-1.2 a $$

Step-by-Step Solution

Answer

The value of \( a \) is 5.3.

Step 1: Isolate the Variable Term

First, we want to isolate the term with the variable on one side of the equation. Start by subtracting 3.36 from both sides of the equation.\[-3 - 3.36 = 3.36 - 1.2a - 3.36\]This simplifies to:\[-6.36 = -1.2a\]

Step 2: Solve for the Variable

Now, to solve for \( a \), divide both sides by -1.2 to get:\[ a = \frac{-6.36}{-1.2} \]This results in:\[ a = 5.3 \]

Key Concepts

Solving Linear Equations

Linear equations are fundamental in algebra and serve as a stepping stone to more complex mathematics. They involve finding the value of an unknown variable that makes the equation true. Typically, a linear equation is in the form of \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. The goal is to determine the value of \(x\) by performing operations that eventually simplify the equation. These operations include addition, subtraction, multiplication, and division, which can change the form of the equation without altering its equality. Understanding linear equations is critical because they can be used to solve real-life problems involving rates, budgets, and unknown quantities. When solving them, always remember that whatever operation you perform on one side of the equation, you must do the same to the other side to maintain balance. This consistency is key in preserving the equality.

Isolation of Variable

The process of isolating a variable means separating it from other numbers or variables in an equation. This step is crucial before finding its value. In an equation like \(-3 = 3.36 - 1.2a\), the first thing to do is to get \(a\) by itself on one side of the equation. Here's how you can do that:

• Identify the term with the variable, in this case, \(-1.2a\).
• Subtract or add numbers to eliminate other terms in the equation. For instance, subtract \(3.36\) from \(3.36 - 1.2a\).

Once you isolate the variable term, all that remains is solving for the variable, which may involve additional operations like division or multiplication. Properly isolating the variable is crucial for ensuring that you find the correct solution.

Step-by-Step Solution

A step-by-step solution is a detailed breakdown of the process to solve a problem, which can be very helpful for understanding the logic behind each move.Let's see how it's done for the equation \(-3 = 3.36 - 1.2a\):

• Step 1: Isolate the variable term by subtracting \(3.36\) from both sides: \(-3 - 3.36 = 3.36 - 1.2a - 3.36\) simplifies to \(-6.36 = -1.2a\).
• Step 2: Solve for \(a\) by dividing both sides by \(-1.2\): \(a = \frac{-6.36}{-1.2}\), which calculates to \(a = 5.3\).

By breaking down each step, we can see exactly how the operations lead us to the solution. This method not only strengthens comprehension but also builds problem-solving confidence in algebra. Each step ensures clarity and provides a structure that can be applied to similar problems.