Problem 7 - Prealgebra

Question

Solve each equation. $$-4 y=1.4$$

Step-by-Step Solution

Answer

The solution to the equation \(-4y = 1.4\) is \(y = -0.35\).

Step 1: Understand the Equation

The equation given is \(-4y = 1.4\). This is a simple linear equation where \(y\) is multiplied by \(-4\) to equal \(1.4\). Our goal is to solve for the variable \(y\).

Step 2: Isolate the Variable

To isolate \(y\), we need to divide both sides of the equation by \(-4\). This means we perform the operation: \(\frac{-4y}{-4} = \frac{1.4}{-4}\).

Step 3: Simplify the Equation

Simplifying both sides, we get: \(y = \frac{1.4}{-4}\). Calculate \(\frac{1.4}{-4}\), which equals \(-0.35\).

Step 4: Verify the Solution

Substitute \(y = -0.35\) back into the original equation to check: \(-4(-0.35) = 1.4\). This simplifies to \(1.4 = 1.4\), confirming that the solution is correct.

Key Concepts

Isolating Variables

Isolating a variable is one of the foundational steps in solving linear equations. It's like peeling away layers to find what's underneath. Our main goal is to "free" the variable on one side so we can determine its value. In the given equation \( -4y = 1.4 \), \( y \) is being multiplied by \( -4 \). To isolate \( y \), we need to do the opposite operation of what's currently affecting it, which is division. This requires us to divide both sides by \( -4 \). Here's the process visually: \( \frac{-4y}{-4} = \frac{1.4}{-4} \). By dividing, we are essentially undoing the multiplication with \( -4 \), leaving us with just \( y \). This process of isolating \( y \) ensures that we only have one term on one side, making it much easier to solve.

Simplifying Equations

Once we've isolated the variable, our next goal is to simplify the equation to find the value. Simplifying makes equations easier to understand and solve. After isolation, our equation becomes \( y = \frac{1.4}{-4} \). We perform the division \( \frac{1.4}{-4} \) to simplify and find the exact value of \( y \). Therefore, \( y = -0.35 \). Simplification involves basic arithmetic to make complex-looking equations clearer and to the point. This reduced form is where we can identify the term for \( y \) precisely. It confirms that simplification transforms the isolated version into a straightforward answer.

Verification of Solutions

Verification of solutions is a crucial step in ensuring the accuracy of your answer. Once you believe you've solved the equation, double-checking is vital. Here, you substitute the obtained solution back into the original equation. In our case, let's substitute \( y = -0.35 \) into \( -4y = 1.4 \). This becomes \( -4(-0.35) = 1.4 \). Calculate this to \( 1.4 = 1.4 \), which confirms our solution is correct. Verification helps confirm that no errors were made during isolation or simplification, solidifying our confidence in the solution. It's like proofreading your math to spot any errors early. This step guarantees that you've considered every possibility and provides peace of mind in your final answer.