Problem 23 - Introductory Algebra for College Students

Question

Solve each equation in using the multiplication property of equality. Be sure to check your proposed $$-51=-y$$

Step-by-Step Solution

Answer

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

Step 1: Interpret the negative sign as multiplication by -1

First, rewrite the equation interpreting the negative sign in front of the \(y\) as multiplication by \(-1\). So the equation \(-51=-y\) can be rewritten as \(-51 = -1 \cdot y\).

Step 2: Use the multiplication property of equality to solve for y

Next, use the multiplication property of equality to solve for \(y\). To do this, we can divide both sides of the equation by \(-1\), which will isolate \(y\). Doing so gives us \(y = 51\).

Step 3: Check answer

To ensure that \(y = 51\) is indeed a solution to the original equation, substitute \(51\) for \(y\) in the original equation: \(-51 = -(51)\). Doing so, we see that \(-51 = -51\), which verifies that \(y = 51\) is a solution.

Key Concepts

Solving Equations

When faced with equations in algebra, our goal is to isolate the variable and find its value. This process involves using various properties to systematically solve for the unknown value. The equation from our example is \(-51 = -y\). To solve this, think of the negative sign in front of the variable \(y\) as being multiplied by \(-1\). This changes the equation to \(-51 = -1 \cdot y\).

The multiplication property of equality states that if you multiply both sides of an equation by the same non-zero number, the equality remains true. In our case, to isolate \(y\), we need to divide both sides by \(-1\), giving us \(y = 51\). This step-by-step approach helps to clearly identify what operations are necessary to solve for \(y\). Remember, keeping equations balanced is key; whatever operation you do to one side, must also be done to the other.

Negative Numbers in Algebra

Negative numbers can seem tricky at first, but they follow specific rules that simplify computations. When dealing with equations like \(-51 = -y\), understanding how negative numbers operate is important. In algebra, a negative sign before a variable means it is being multiplied by \(-1\). So, \(-y\) is the same as \(-1 \cdot y\).

To solve these equations, we either multiply or divide by \(-1\) to change the sign and isolate the variable. Here's a handy tip:

• Multiplying or dividing by a negative flips the sign.
• Be cautious about the order of operations and apply operations throughout the whole equation consistently.

Recognizing these principles allows you to solve equations with negative numbers confidently.

Checking Solutions

After solving an equation, it's crucial to check if your answer is correct. This is an easy way to verify that no mistakes were made during calculations. To check the solution \(y = 51\) for the equation \(-51 = -y\), substitute \(51\) back into the original equation.

So, we replace \(y\) with \(51\), resulting in \(-51 = -(51)\). Simplifying, we see \(-51 = -51\), which confirms the balance is true, affirming our solution.

Always check solutions by substituting back into the original equation. This step ensures accuracy and builds confidence in your solving skills.