Problem 70 - Elementary Algebra

Question

Solve. $$ 0=-3.1 a+32.55 $$

Step-by-Step Solution

Answer

The solution is \( a = 10.5 \).

Step 1: Isolate the Variable

To solve for the variable \( a \), we need to isolate it on one side of the equation. The equation is given as \( 0 = -3.1a + 32.55 \). To isolate \( a \), first subtract 32.55 from both sides of the equation to move the constant term to the left side. This gives us \( -32.55 = -3.1a \).

Step 2: Solve for the Variable

After isolating the term \(-3.1a\), divide both sides by \(-3.1\) to solve for \( a \). This yields \( a = \frac{-32.55}{-3.1} \). Calculating this expression gives us \( a = 10.5 \).

Key Concepts

Isolating the Variable

When solving an equation, a crucial step is to isolate the variable. This means you want to get the variable alone on one side of the equation. In the equation \(0 = -3.1a + 32.55\), our variable is \(a\). To isolate \(a\), you need to remove any numbers or terms that are with \(a\) on the same side. In this case, you can subtract 32.55 from both sides of the equation.

This subtraction helps in maintaining the balance of the equation. Remember, whatever you do to one side of an equation, you must do to the other, to keep it balanced. After subtracting 32.55 from both sides, you get \(-32.55 = -3.1a\). Now, \(a\) is alone on one side with a coefficient, making it easier to solve in the next steps.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to simplify and solve equations. Once you have isolated the variable with its coefficient, the next step is to make the coefficient one. In our equation, we have \(-3.1a\) as a term on the right after isolation. The goal is to make \(a\) standalone without any coefficients or terms attached.

To achieve this, divide both sides of the equation by the coefficient of the variable. Here, you divide both sides by \(-3.1\). This gives us:

• \(\frac{-32.55}{-3.1} = a\)

Dividing by the negative flips the sign, so the calculation simplifies correctly, resulting in \(a = 10.5\). This operation scales back the equation to its simplest form, showing the precise value of the variable.

Equation Solving Steps

Solving equations can be straightforward if you follow logical steps. Here’s a simple guide to use each time:

1. **Identify and isolate the variable**: Begin by figuring out which term includes your variable, then perform operations to get the variable by itself on one side of the equation.2. **Use algebraic manipulation**: If there is a coefficient with your variable, use division or multiplication to make it one.3. **Solve and check**: Once your variable is isolated and the coefficient is one, compute the remaining arithmetic to find the variable's value. In our case, this resulted in \(a = 10.5\).

Remember, it’s always a good practice to plug the solution back into the original equation to verify your result. This cross-check ensures your solution's accuracy and helps reinforce your understanding of the entire process.