Problem 9 - Prealgebra

Question

Solve each equation. $$0.5 n=-0.4$$

Step-by-Step Solution

Answer

The solution is \(n = -0.8\).

Step 1: Multiply both sides by 2

The equation is given as \(0.5n = -0.4\). To eliminate the decimal coefficient of \(n\), we can multiply every term of the equation by 2. Multiplying both sides by 2 gives us: \(2 imes 0.5n = 2 imes -0.4\), which simplifies to \(n = -0.8\).

Step 2: Check the solution

To ensure our solution is correct, substitute \(n = -0.8\) back into the original equation: \(0.5(-0.8) = -0.4\). Calculate the left-hand side: \(0.5 imes (-0.8) = -0.4\), which matches the right-hand side. Therefore, the solution \(n = -0.8\) is verified.

Key Concepts

Decimal Coefficients

In algebra, you often come across equations with decimal coefficients which can seem a bit daunting at first. A decimal coefficient is simply a number less than one that multiplies a variable, like 0.5 in the equation \(0.5n = -0.4\). Decimals can make calculations appear tricky, but they can simplify equations by replacing fractions or complex numbers in some scenarios.
To simplify equations with decimal coefficients, a go-to method is to multiply all terms by a power of ten that will eliminate the decimal. This makes equations easier to handle. For example, in our exercise, we multiplied every term by 2 to get rid of the decimal coefficient of \(n\). After doing that, you're left with a simpler equation that is as straightforward as handling whole numbers.

Multiplication Property of Equality

The multiplication property of equality is a fundamental principle in algebra. According to this property, if you multiply both sides of an equation by the same nonzero number, the equation's equality is preserved. This is because whatever operation you perform on one side must also be done on the other to keep the equation balanced.
In the original exercise, the equation \(0.5n = -0.4\) was simplified by utilizing this property. We multiplied both sides by 2, which eliminated the decimal and offered \(n = -0.8\). The multiplication property helped us maintain the equation's validity and arrive at a clearer, more solvable equation.

Verification of Solutions

Once you've found a potential solution for an equation, it's always wise to verify it. Verification involves substituting the solution back into the original equation to ensure both sides of the equation remain equal. This step confirms that the solution is correct and consistent with the given equation.
In our example, after solving for \(n\) as \(-0.8\), substituting it back into \(0.5n = -0.4\) lets us verify the calculation. Multiplying \(0.5\) by \(-0.8\) gives us \(-0.4\), which matches the original equation's right-hand side. This verification step confirms that \(n = -0.8\) is indeed the correct solution, guaranteeing you haven't made any arithmetic errors along the way.