Problem 15 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the given equation

The given equation is \(-16y=0\)

Step 2: Isolate the variable

To isolate the variable \(y\), divide both sides of the equation by -16. \(-16y/-16 = 0/-16\)

Step 3: Simplify the resulting expression

Upon simplifying, you get \(y=0\).

Step 4: Check the solution

Substitute the obtained value of \(y\)=0 into the original equation to confirm whether it holds true. The equation becomes \(-16*0= 0\). Hence, \(y=0\) is indeed the solution.

Key Concepts

Solving Equations

When tackling equations, the goal is to find the value of the unknown variable that makes the equation true. This is the essence of solving equations. In our example, we are given the equation \(-16y = 0\). Our task is to determine what value of \(y\) will satisfy this equation.

The first step in solving any equation is identifying the elements involved: the operations and the terms. Here, \(-16y\) is the term, which means \(y\) is multiplied by \(-16\). So, to "solve" for \(y\), we need to undo this multiplication using a strategy known as algebraic manipulation, which is explored in the next section.

It's crucial to approach these problems methodically, to break them down into understandable steps to confidently reach the solution.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging, simplifying, or performing operations on equations to make them easier to work with. It’s a fundamental skill in solving equations. In the given equation, \(-16y = 0\), we use the multiplication property of equality to isolate the variable \(y\).

• First, recognize that \(y\) is being multiplied by \(-16\). The inverse operation of multiplication is division.
• Apply division to both sides of the equation by \(-16\) to maintain equality. This gives us: \[-16y / -16 = 0 / -16\]
• By dividing, the \(-16\)s on the left-hand side cancel out, and we simplify the right side to get \(y = 0\).

The beauty of algebraic manipulation lies in its ability to transform complex equations into more straightforward forms, making it easier to identify the solutions.

Equation Checking

Once you have found a solution to an equation, it's always important to check your work. Equation checking involves substituting the value back into the original equation to ensure that it makes the equation true. This step helps to confirm that no mistakes were made during your calculations.

In our example, after determining that \(y = 0\), we substitute \(y\) back into the original equation \(-16y = 0\). When we do this, the equation becomes \(-16 \times 0 = 0\).

Since both sides of the equation are equal, this verifies that our solution is correct. Checking your work might seem like a redundant step, but it’s a critical part of problem-solving. This practice ensures accuracy and builds confidence in your mathematical reasoning.