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Chapter 2: Problem 88

Use the multiplication or division property of equality to solve each equation. $$-0.2 y=51$$

Short Answer

Expert verified
The solution of the equation is \( y = -255 \)

Step by step solution

01

Multiply by the reciprocal

To isolate \(y\), the first step is to multiply both sides of the equation by the reciprocal of the coefficient of \(y\). The coefficient is -0.2, so the reciprocal would be \(-\frac{1}{0.2}\) or -5. Hence, the equation becomes:\(-5 \times -0.2y = -5 \times 51\) which simplifies to \( y = -255 \)

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Key Concepts

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

Multiplication Property of Equality
The multiplication property of equality is a fundamental rule in solving equations. It states that if you multiply both sides of an equation by the same number, the equation remains balanced. This means whatever operation you do to one side, you do to the other as well.

For example, if you have the equation \( -0.2y = 51 \), you can multiply both sides by a number to maintain balance. In equation solving, this often involves multiplying by the reciprocal of a coefficient to isolate the variable. Keeping equations balanced in this way ensures that the equality holds true throughout the steps.
Division Property of Equality
The division property of equality is another key concept that keeps equations balanced. This principle states that if you divide both sides of an equation by the same nonzero number, the equality is maintained. It is the counterpart to the multiplication property.

Dividing is particularly useful when the coefficient of a variable is an integer or a simple fraction. But in our specific exercise example, multiplication with a reciprocal was a more suitable step. Still, understanding both properties allows flexibility in problem-solving.
Reciprocal
The reciprocal of a number is simply the inverse of that number. When you multiply a number by its reciprocal, the result is 1. For example, the reciprocal of \(-0.2\) can be found by flipping it. Rewriting \(-0.2\) as a fraction gives us \(-\frac{1}{5}\). The reciprocal of \(-0.2\) or \(-\frac{1}{5}\) is \(-5\).

In solving the equation \(-0.2y = 51\), multiplying by the reciprocal of \(-0.2\) allowed us to effectively isolate \(y\), turning \(-0.2y\) into \(y\). This method is powerful for quickly simplifying terms to solve for variables.
Coefficients
A coefficient is the number that is multiplied by the variable in an algebraic expression. In the equation \(-0.2y = 51\), \(-0.2\) is known as the coefficient of \(y\).

Coefficients give value to variables and determine how much weight they carry within an equation. Coefficients can be whole numbers, decimals, or fractions. Recognizing them in an equation is crucial as they dictate the manipulation steps, such as finding reciprocals to solve for the variable. By isolating the variable term from the coefficient, we can solve the equation effectively.

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