Problem 27 - Prealgebra

Question

Solve each equation. Check your solution. $$-0.4 y=2$$

Step-by-Step Solution

Answer

The solution is \(y = -5\).

Step 1: Isolate the Variable

To solve the equation \(-0.4y = 2\), start by isolating the variable "\(y\)". To do this, divide both sides of the equation by \(-0.4\): \(\frac{-0.4y}{-0.4} = \frac{2}{-0.4}\)This simplifies to \(y = -5\).

Step 2: Verify the Solution

To ensure the solution is correct, substitute \(y = -5\) back into the original equation:\(-0.4(-5) = 2\).Calculate the left side: \(-0.4 imes -5 = 2\).Since the left and right sides of the equation are equal, the solution \(y = -5\) is verified.

Key Concepts

Solving Equations

When you're faced with an equation like \(-0.4y = 2\), you're tasked with finding the value of the variable, in this case, \(y\), that makes the equation true.The main goal is to transform the equation step by step until the variable stands alone on one side of the equation.This transformation process is what we refer to as 'solving an equation'.

• Start by analyzing the given equation. In our case, we have a simple linear equation \(-0.4y = 2\).
• Apply arithmetic operations that modify the equation while keeping it balanced and true.
• It may involve adding, subtracting, multiplying, or dividing both sides of the equation equally.

By carefully manipulating the equation, you can simplify it to eventually find the value of \(y\).

Isolate the Variable

The key to solving any linear equation is to 'isolate the variable', which means getting the variable by itself on one side of the equation.In our example, to isolate \(y\) in the equation \(-0.4y = 2\), you have to perform the opposite operation to what is currently being done to \(y\).

• Since \(y\) is being multiplied by \(-0.4\), you will want to divide both sides of the equation by \(-0.4\).
• The division simplifies the equation, leaving \(y\) standing alone: \[ \frac{-0.4y}{-0.4} = \frac{2}{-0.4} \]
• This calculates to \(y = -5\), giving you the solution to the equation.

Isolating the variable is a crucial step because it simplifies the problem, making it manageable and easily solvable.

Verification of Solutions

Verification is the final but vital step in solving equations; it ensures that you've found the right solution.After calculating \(y = -5\), you need to check if this value satisfies the original equation \(-0.4y = 2\).

• Replace the variable \(y\) with the found value in the original equation.
• Substitute \(y = -5\) back in: \[ -0.4(-5) = 2 \]
• Calculate the left side: \( -0.4 \times -5 \) equals \(2\).

If both sides of the equation are equal, which they are, then \(y = -5\) is verified as the correct solution.Verification confirms accuracy and provides you with confidence that the work was done correctly.