Problem 15 - Prealgebra

Question

Solve each equation. Check your solution. $$4.7=a+7.1$$

Step-by-Step Solution

Answer

The solution is \( a = -2.4 \). It checks out because substituting back verifies the equation.

Step 1: Identify Equation

First, we identify the equation we need to solve. The equation given is \( 4.7 = a + 7.1 \).

Step 2: Isolate the Variable

To solve for \( a \), we need to isolate the variable. We do this by subtracting 7.1 from both sides of the equation:\[4.7 - 7.1 = a + 7.1 - 7.1\]Simplify the equation:\[4.7 - 7.1 = a\]

Step 3: Simplify the Equation

Now, subtract 7.1 from 4.7:\[4.7 - 7.1 = -2.4\]So the equation becomes \( a = -2.4 \).

Step 4: Check the Solution

To ensure the solution is correct, substitute \( a = -2.4 \) back into the original equation:\[4.7 = -2.4 + 7.1\]Calculate the right side:\[-2.4 + 7.1 = 4.7\]Since both sides of the equation equal 4.7, the solution is verified.

Key Concepts

Equation Solving

When solving linear equations, it's essential to understand the goal: find the value of the unknown variable that makes the equation true. In our problem, we are given the equation \(4.7 = a + 7.1\). Solving equations involves manipulating them to isolate the variable. This process requires practicing balancing both sides by applying the same operation to each side. This ensures equality is maintained, guiding us to the solution.

Here’s a simple guide to solve linear equations:

• Identify the equation you need to work with.
• Determine steps to isolate the variable.
• Simplify expressions as needed.
• Check your solution by substituting it back into the original equation.

By following these steps, we can systematically approach solving any linear equation with confidence.

Isolating Variables

Isolating the variable is all about getting the variable you're solving for, by itself, on one side of the equation. This often involves undoing the operations that have been applied to this variable. In our example \(4.7 = a + 7.1\), we want to solve for \(a\).

To do this, observe the equation closely:

• The variable \(a\) is being added to 7.1.
• To isolate \(a\), you perform the opposite operation of what's currently happening.

In this case, subtract 7.1 from both sides. This allows you to cancel out the +7.1 on the right, leaving \(a\) by itself. This results in a simpler equation: \(a = -2.4\). Notice how essential understanding the operations is in gaining the isolation successfully.

Checking Solutions

After isolating the variable, it's crucial to check if the solution is correct. This step ensures that we made no mistakes in our calculations or simplifications. For the equation \(4.7 = a + 7.1\), and with our found solution \(a = -2.4\), we substitute back to verify:

• Replace \(a\) with \(-2.4\) in the equation: \(4.7 = -2.4 + 7.1\).
• Calculate the right-hand side: \(-2.4 + 7.1 = 4.7\).

The calculations show both sides of the equation are equal, confirming our solution is correct. Checking helps you feel confident in your results and reinforces your understanding of equation-solving processes.

Simplifying Expressions

Simplifying expressions makes equations easier to work with by reducing complexity. In our problem, after isolating \(a\), we simplified the expression \(4.7 - 7.1\). Simplification involves performing basic arithmetic or combining like terms when possible.

• Start with identifying numeric terms that can be simplified on each side of the equation.
• Perform the arithmetic operation: \(4.7 - 7.1 = -2.4\).
• This results in a simpler concept representation enabling clearer inspection: \(a = -2.4\).

Subtle simplifications often reveal more straightforward paths to the solution, aiding comprehension and reducing errors along the way.