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Chapter 1: Problem 11

Solve each equation. Check your solution. $$ 1.8 a-5=-2.3 $$

Short Answer

Expert verified
The solution is \(a = 1.5\).

Step by step solution

01

Isolate the Variable Term

To solve the equation \(1.8a - 5 = -2.3\), start by isolating the term with the variable \(a\). Add 5 to both sides of the equation to eliminate the constant term on the left side. This gives:\[1.8a - 5 + 5 = -2.3 + 5\]which simplifies to:\[1.8a = 2.7\]
02

Solve for the Variable

Now that the term with the variable \(1.8a\) is isolated, divide both sides by 1.8 to solve for \(a\):\[\frac{1.8a}{1.8} = \frac{2.7}{1.8}\]This simplifies to:\[a = 1.5\]
03

Verify the Solution

To ensure the solution is correct, substitute \(a = 1.5\) back into the original equation and check that both sides are equal. Substitute as follows:\[1.8(1.5) - 5 = -2.3\]Calculate the left side:\[2.7 - 5 = -2.3\]Simplify to find:\[-2.3 = -2.3\]Since both sides are equal, \(a = 1.5\) is the correct solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the Variable
When solving linear equations, the first step is often isolating the variable. This means getting the variable by itself on one side of the equation. In our example, the equation is \(1.8a - 5 = -2.3\). To isolate \(a\), we need to remove any constants on its side. We do this by performing basic algebra operations. Here, we add 5 to both sides to eliminate the \(-5\) that is with the variable term. Remember, whatever you do to one side of the equation, you must do to the other to maintain balance. After adding 5, we have \(1.8a = 2.7\), which successfully isolates the term \(1.8a\) with the variable.
Verify Solutions
Once you've found a solution, it's important to verify it. This ensures that you've correctly solved the equation without mistakes. Verification involves substituting the found value back into the original equation. In our problem, we solved for \(a = 1.5\). To verify, substitute \(1.5\) back into the equation: \(1.8(1.5) - 5 = -2.3\). Calculate \(1.8 \times 1.5\), which is \(2.7\). Subtract 5 from \(2.7\), and you should get \(-2.3\). Both sides match, confirming that \(a = 1.5\) is the correct solution.
Verification is a useful step to catch any errors in calculations. It is always good practice in math, especially during tests or when you're learning.
Basic Algebra Operations
Basic algebra operations are the foundational rules that guide us in solving equations. These include addition, subtraction, multiplication, and division. They help us manipulate equations to isolate variables or simplify expressions. Let's take our exercise as an example:
  • Addition and subtraction: We added 5 to both sides to eliminate the \(-5\) term. This is an example of using addition to balance the equation.
  • Division: Once we isolated \(1.8a\), we divided both sides by 1.8 to solve for \(a\).
Remember, the key is to perform the same operation on both sides of the equation to keep it balanced. Keeping equations balanced is essential when solving for any variable. Understanding these basic operations makes solving more complex equations possible.

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