Problem 16 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation using the multiplication property of equality. Be sure to check your proposed solutions. $$-16 y=0$$

Step-by-Step Solution

Answer

The solution to the equation \(-16 y = 0\) is \(y = 0\)

Step 1: Solve the Equation

The given equation is \(-16 y = 0\). The multiplication property of equality states that if you multiply both sides of an equation by the same nonzero number, the sides remain equal. To isolate \(y\), divide both sides of the equation by \(-16\). Hence, the equation becomes \(y = 0 / -16 = 0\)

Step 2: Check the Solution

Substitute \(y = 0\) back into the original equation to verify the solution: \(-16 * 0 = 0\). The left side of the equation equals the right side, so the solution is verified.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of the variable that makes the equation true. In this exercise, we were given an equation involving multiplication: -16y = 0. To solve it, we used the multiplication property of equality, which allows us to perform the same operation on both sides of an equation without changing its solution. This property is crucial because it helps keep the equation balanced.
The primary goal when solving equations is to transform the equation into a simpler form so you can find the variable's value. You want to get the variable by itself on one side of the equation while the other side is just a number.
In this exercise, we divided both sides by -16, which gave us: -16y / -16 = 0 / -16. This operation simplified to y = 0, which is our solution.

Isolation of Variables

Isolation of variables is about getting the variable in the equation all by itself. The ultimate aim is to make the equation read something like "y = ...". Achieving this often involves reversing operations that have been applied to the variable.
In the given exercise with the equation -16y = 0, y is being multiplied by -16. To isolate y, we need an operation that undoes multiplication by -16. This is where division comes in handy. By dividing both sides of the equation by -16, we effectively nullify the multiplication and solve for y: -16y / -16 = 0 / -16, which simplifies to: y = 0.
The concept of isolating the variable is critical because it leads us directly to the solution, showing the value that makes the equation correct.

Verification of Solutions

Verification of solutions is a crucial step in ensuring that the value found truly satisfies the original equation. This step is about substituting the solution back into the original equation to check its correctness.
For example, once we found that y = 0 was the solution, we substitute it back into the original equation: -16 * 0 = 0. The left side of the equation equals the right side, confirming that our solution is correct.
Verification is not just a routine step; it’s an essential assurance that the solution makes sense and that no mistakes were made during the solving process. It's like a final proof that what we have found is indeed the correct answer.