Problem 24 - 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. $$-51=-y$$

Step-by-Step Solution

Answer

The solution to the equation \(-51 = -y\) is \(y = 51\).

Step 1: Analyze the equation

First, we need to understand that we have a simple equation which equates \(-51\) to \(-y\). Here, \(y\) is the variable we have to find. The negative sign in front of \(y\) means that the variable \(y\) is multiplied by \(-1\). To find the value of \(y\), we need to make \(y\) the subject of the equation by isolating it on one side.

Step 2: Apply the multiplication property of equality

In order to get \(y\) on one side of the equation, we will multiply both sides of the equation by \(-1\). This operation is performed to keep the balance of the equation. So the modified equation becomes: \(-1 \cdot -51 = -1 \cdot -y\).

Step 3: Calculate the value of y

After multiplying, we get: \(51 = y\). So the variable \(y\) is equal to \(51\).

Step 4: Verify the solution

Lastly, we substitute \(y = 51\) in the original equation to verify the solution. Substituting the value, we get: \(-51 = -(51)\), which is a true statement.

Key Concepts

Solving Linear Equations

When we talk about solving linear equations, we refer to the process of finding the value of the variable that makes the equation true. A linear equation is an equation where the variable has a maximum power of 1. This means that you don't have variables raised to higher powers, such as squared or cubed. In the given example, the linear equation is \(-51 = -y\). This simple form still requires strategic steps to solve it. Here, your main aim is to determine what value of \(y\) will balance both sides of the equation.

• Start by recognizing the signs and numbers: Notice that \(-y\) indicates \(y\) is being multiplied by \(-1\).
• Use basic arithmetic rules: You will need to perform the same operation on both sides to maintain equal balance in the equation.

Solving linear equations can always be broken down into clear, manageable steps. Remember, the key is to perform operations that simplify the variables present in the equation.

Isolation of Variables

Isolation of variables is a crucial step in solving equations. It involves manipulating the equation to get the variable on one side by itself. Think of it as trying to get the variable alone so that you can see its exact value.
In the equation \(-51 = -y\), the variable we need to isolate is \(y\). Since \(y\) is multiplied by \(-1\), we must use a mathematical operation that cancels out this multiplication.

• Use the multiplication property of equality: Multiply both sides of the equation by \(-1\). This helps counteract the multiplying \(-1\) by \(y\).
• Keep the equation balanced: Whatever action you do on one side, you must do the same on the other side. By doing so, the equation remains true.

After performing the operation, the variable stands alone as \(y = 51\). Now, the equation is simplified, and you can clearly see the value of \(y\). Successfully isolating the variable is essential to solving any equation.

Verifying Solutions

Verifying solutions is the final, crucial step in ensuring that the value found for the variable truly satisfies the original equation. This step validates your solution and reassures you that the operations you performed were correct.
For instance, after finding \(y = 51\) from the equation \(-51 = -y\), substitute \(51\) back into the original equation to double-check.
Put \(-51\) on one side and substitute \(-51\) for \(-(-51)\), which simplifies to \(-51 = -51\).

• Double-check your work: Always go back to the original equation and insert the found value.
• Look for matching sides: Both sides should be equal once the substitution is made; if they match, you have correctly solved the equation!

This step is always important, no matter how confident you are in your algebraic skills. Ensuring accuracy helps prevent errors and solidifies your understanding of solving linear equations. Be diligent in verifying to cultivate a habit of precision in math.