Problem 15 - Prealgebra

Question

Use the multiplication property of equality to solve each of the following equations. In each case, show all the steps. $$5 x=-35$$

Step-by-Step Solution

Answer

The solution to the equation is \(-7\).

Step 1: Identify the Property

The multiplication property of equality states that if you multiply both sides of an equation by the same nonzero number, the equality is maintained. For this equation, we'll use a specific application of this property: instead of multiplying the equation to solve, we actually divide both sides by the coefficient of the variable.

Step 2: Divide Both Sides

Divide both sides of the equation by 5 to isolate the variable, since the equation is currently 5 times the variable. So we have: \[\frac{5x}{5} = \frac{-35}{5}\]

Step 3: Simplify the Equation

Simplify both sides of the equation. The left-hand side simplifies to \( x \), and the right-hand side simplifies to \( -7 \). Thus, the equation simplifies to: \[ x = -7 \]

Key Concepts

Multiplication Property of Equality

The multiplication property of equality is a fundamental rule in mathematics that helps us solve equations by keeping them balanced. An equation can be thought of as a set of scales, with a balance of weights on both sides. This property states that if you multiply (or divide) both sides of an equation by the same nonzero number, the equality will still hold true. It's a simple but powerful tool for solving equations efficiently.

When dealing with equations, this property allows us to modify both sides to make the problem simpler, while maintaining the essential mathematical relationship.

Here's what you need to remember about this property:

• It works for multiplication and division, as long as the number used is not zero.
• This property is often used in tandem with other operations to isolate the variable.
• In our exercise, though we say we use the multiplication property, we're technically dividing because the equation involves multiplication to begin with.

Solving Equations

Solving equations is about finding the value of the variable that makes the equation true. In our example, the equation is \(5x = -35\). To solve this, we need to perform operations that get \(x\) by itself.

The process generally involves combining like terms and simplifying both sides. Sometimes you add or subtract numbers to move them from one side of the equation to the other, and sometimes you multiply or divide both sides, like in our example.

Key steps in solving equations include:

• Identifying the operations needed to isolate the variable.
• Applying these operations systematically to both sides.
• Checking your solution by substituting the variable back into the original equation to see if it results in a true statement.

Remember, maintaining balance is crucial, so any action you take on one side must be mirrored on the other.

Variable Isolation

The goal of variable isolation is to have the variable all by itself on one side of the equation. This helps you determine what the variable equals, hence solving the equation.

In prealgebra, isolating the variable is often done using properties of equality. In our case, we wanted to get \(x\) by itself, starting from \(5x = -35\). We used division, because \(5x\) indicates 5 multiplied by \(x\). Dividing both sides by 5 leaves us with \(x\) on one side of the equation: \(x = \frac{-35}{5}\). Simplifying gives us \(x = -7\).

Tips for successful variable isolation:

• Perform the same operation on both sides to maintain the equation's balance.
• Simplify the equation step by step to avoid mistakes.
• Always check your work by plugging the isolated variable back into the original equation.

By following these steps, isolating the variable can become a methodical and stress-free process.