Problem 19 - Prealgebra

Question

Solve each equation. $$3 a+4.6=7 a+5.3$$

Step-by-Step Solution

Answer

The solution is \(a = -0.175\).

Step 1: Isolate the Variable Terms on One Side

Start by rearranging the equation to get the variable terms on one side. Subtract \(3a\) from both sides to keep the variables on the right side.\[3a + 4.6 - 3a = 7a + 5.3 - 3a\]This simplifies to:\[4.6 = 4a + 5.3\]

Step 2: Remove Constant from Variable's Side

Eliminate the constant from the side with the variable by subtracting 5.3 from both sides:\[4.6 - 5.3 = 4a + 5.3 - 5.3\]This simplifies to:\[-0.7 = 4a\]

Step 3: Solve for the Variable

Finally, solve for \(a\) by dividing both sides by 4:\[\frac{-0.7}{4} = \frac{4a}{4}\]Thus, we find:\[a = -0.175\]

Key Concepts

Variable Isolation

In the world of linear equations, isolating the variable is one of the first and crucial steps. The goal is to get the variable by itself on one side of the equation. This makes it easier to analyze and solve the equation.
To achieve isolation, one common tactic is to eliminate the variable term on one side by using the inverse operation. For example, if the equation reads \(3a + 4.6 = 7a + 5.3\), you want to move all the terms containing \(a\) to one side. You do this by subtracting \(3a\) from both sides:

• On the left side, \(3a\) cancels out, leaving just the constant.
• On the right side, you subtract \(3a\) from \(7a\), simplifying the equation.

Now, you've effectively moved all the \(a\) terms to one side, which is a great start towards solving the equation.

Simplifying Equations

Once you've moved the variable terms to one side, the next logical step is simplifying the equation. Simplification is about reducing the equation to its simplest form to allow for straightforward solutions.
In our example, after isolating the variable on one side, the equation was transformed to \(4.6 = 4a + 5.3\). The next job is to simplify further by removing any constants from the side with the variable.
This involves eliminating 5.3 on the right side by subtracting it from both sides. The calculation looks like this:

• Subtract 5.3 from both sides: \(4.6 - 5.3 = 4a + 5.3 - 5.3\)
• The result simplifies to \(-0.7 = 4a\)

Now, you're left with a more straightforward equation that directly links the variable and a number.

Solving for Variables

The final step in addressing linear equations is solving for the variable. When the equation is simplified, as in our case with \(-0.7 = 4a\), the path to finding the value of \(a\) is clear. The objective now is to isolate \(a\) completely.

• You achieve this by dividing both sides of the equation by the coefficient of \(a\), in this case, 4.
• So, \(\frac{-0.7}{4} = \frac{4a}{4}\).

The division neatly cancels out the 4 on the right side, resulting in \(a = -0.175\). This provides you with the value of the variable, thus solving the equation entirely. Each step ensures that the equation remains balanced, always performing the same operation on both sides.