Problem 25 - Introductory Algebra for College Students

Question

Solve each equation in using the multiplication property of equality. Be sure to check your proposed $$-\frac{x}{5}=-1$$

Step-by-Step Solution

Answer

The solution to the equation \( -\frac{x}{5} = -1 \) is \( x = 5 \).

Step 1: Set Up the Equation

Write the given equation \( -\frac{x}{5} = -1 \).

Step 2: Use the multiplication property of equality

Multiply both sides of the equation by -5 to isolate x on the left side. This gives you \( x = -1 \times -5 \). The multiplication property of equality allows us to multiply both sides by the same value (in this case -5) making sure the balance of the equation is maintained.

Step 3: Calculate the Result

Multiply -1 by -5 to get \( x = 5 \). So, the solution is \( x = 5 \).

Step 4: Checking the Solution

Substitute \( x = 5 \) into the original equation \( -\frac{x}{5} = -1 \), which gives \( -\frac{5}{5} = -1 \). Since -1 = -1, the solution checks.

Key Concepts

Solving Equations

In the journey of learning algebra, solving equations is like unraveling a mystery by finding the hidden value of a variable that makes an equation equal on both sides. When you're faced with an equation like \( -\frac{x}{5} = -1 \), your task is to determine what value of \( x \) makes that equation true.
The steps to solve such equations can be boiled down to a few clear actions:

• Identify the equation and the variable that you want to isolate.
• Use algebraic techniques to manipulate the equation, with the goal of isolating the variable on one side.
• Finally, verify that your found value satisfies the original equation.

For example, by multiplying both sides of \( -\frac{x}{5} = -1 \) by -5, the variable \( x \) becomes isolated, and the equation is simplified to \( x = 5 \). Verifying your solution ensures that no errors were made during manipulation.

Properties of Equality

The properties of equality are the rules that dictate how you manipulate an equation without changing its overall balance. One crucial property is the multiplication property of equality, used prominently in solving \( -\frac{x}{5} = -1 \).
**Core Properties Include:**

• Addition Property: If you add the same number to both sides of an equation, the equality holds true.
• Subtraction Property: Similar to addition, subtracting the same number from both sides maintains equality.
• Multiplication Property: By multiplying both sides by the same non-zero number, the balance of the equation remains.
• Division Property: Dividing both sides by the same non-zero number keeps the equality valid.

In our example, the multiplication property of equality by multiplying both sides of \( -\frac{x}{5} = -1 \) by -5 results in \( x = 5 \). This manipulation keeps the equation's balance, allowing us to solve it correctly.

Algebra Concepts

Understanding algebra requires grasping foundational concepts that form the basis of manipulating expressions and solving equations. These concepts are not just rules to memorize but tools to use, allowing you to troubleshoot and solve problems effectively.
Here's a quick rundown of key algebra concepts that are often applied:

• Variables: Symbols like \( x \) or \( y \) represent unknown numbers and can be manipulated to find these unknowns.
• Coefficients: These are numbers directly multiplied by variables, such as the -1 in \( -1x \).
• Balancing Equations: Always aim to keep equations balanced, meaning what's done on one side should be done equally on the other.
• Simplification: Combining like terms and using properties of equality to simplify expressions and equations.

In the exercise with \( -\frac{x}{5} = -1 \), each of these algebra concepts plays a role, illustrating how algebra is a mix of strategy, rules, and the art of problem-solving.