Problem 23 - 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. $$-47=-y$$

Step-by-Step Solution

Answer

The solution to the equation \(-47 = -y\) is \(y = 47\).

Step 1: Identify the variable to solve for

In this case, the variable we need to solve for is \(y\).

Step 2: Apply the multiplication property of equality

Multiply both sides of the equation by -1. The multiplication property of equality tells us that we can do this without changing the equality. So, \((-1) \times -47 = (-1) \times -y\). Multiply out each side to get \(47 = y\).

Step 3: Check the solution

Substitute the solution back into the original equation to check if it's correct. In the original equation, replace \(y\) with 47 and simplify: \(-47 = -47\). The equation is true, which tells us that \(y = 47\) is the correct solution.

Key Concepts

Solving Equations

When solving equations, the goal is to find the value of the unknown variable that makes the equation true. This involves manipulating the equation in a way that isolates the variable on one side. In our case, the variable is \( y \) in the equation \( -47 = -y \).

To solve such equations, you can use properties of equality like the multiplication property. This property states that if you multiply or divide both sides of an equation by the same non-zero number, the equality will still hold true.

Let's consider the equation \( -47 = -y \). To make the variable \( y \) positive, you'll multiply both sides by \(-1\). This adjustment simplifies the calculation and ensures you maintain the equation's balance.

Check Solutions

Checking your solution is a vital step in the process of solving equations. Once you propose a solution, you should substitute it back into the original equation to verify its correctness.

For example, when we found that \( y = 47 \) satisfies the equation \( -47 = -y \), we substituted 47 back into the equation as \( -47 = -47 \). Since the two sides of the equation are equal, this confirmed that our solution was correct.

In practice, checking ensures not only accuracy but also builds confidence in the method used to find the solution. Remember, if the equation doesn't balance, revisit your steps to find and correct any mistakes.

Variable Isolation

Variable isolation is a technique used to solve for the unknown in an equation. It involves rearranging the equation by performing operations that simplify the expression until the variable stands alone on one side of the equation.

In the exercise given, you have \( -47 = -y \). To isolate \( y \), apply the multiplication property of equality, which in this context means multiplying both sides of the equation by \(-1\).

This operation changes the equation from \( -47 = -y \) to \( 47 = y \), effectively isolating the variable.

• Step 1: Identify the variable to be isolated, which is \( y \).
• Step 2: Decide on an operation that will simplify the expression around \( y \).
• Step 3: Perform the operation equally on both sides of the equation.

By isolating the variable, you make it possible to directly determine its value, ensuring a clear path to the solution.