Problem 24 - Introductory Algebra for College Students

Question

Solve each equation in using the multiplication property of equality. Be sure to check your proposed $$-\frac{x}{5}=-9$$

Step-by-Step Solution

Answer

The solution to the equation \(-\frac{x}{5} = -9\) is \(x = 45\).

Step 1: Applying Multiplication Property of Equality

The goal is to isolate the variable \(x\) by eliminating the fraction. The multiplication property of equality can be used to achieve this. It states that if \(a = b\), then \(a * c = b * c\). Therefore, multiply both sides of the equation \(-\frac{x}{5} = -9\) by -5.

Step 2: Simplify the Equation

After applying the multiplication property, the equation simplifies to \(x = 45\).

Step 3: Checking the Solution

Substitute \(x = 45\) into the original equation \(-\frac{x}{5} = -9\) and check if both sides are equal. Which results in \(-\frac{45}{5} = -9\), which simplifies to \(-9 = -9\), a true statement. Hence \(x = 45\) is the correct solution.

Key Concepts

Solving Linear Equations

Linear equations are foundational in algebra and involve finding the value of variables that make the equation true. These equations take on the form of a straight line when graphed, hence the name 'linear'. To solve them, one must perform operations that preserve the equation's balance, maintaining the equal value on both sides.

Strategies to solve linear equations include simplifying the expression, using inverse operations, and systematically moving terms from one side of the equation to the other to isolate the variable. Remember that whatever operation you do to one side of the equation, you must do to the other to maintain equality. This concept ensures that the linear equation remains balanced through all manipulations.

Algebraic Fractions

Algebraic fractions are fractions that contain variables in their numerators, denominators, or both. They operate under the same principles as numeric fractions, but they require additional steps, such as factoring, finding common denominators, and simplifying.

When you encounter algebraic fractions in equations, particularly those that involve the variable you're solving for in the denominator, you need to carefully manage these terms. Use multiplication or division to clear the fractions and reduce the equation to a simpler form, often converting them into integers or whole number coefficients that are easier to handle.

Isolating Variables

Isolating the variable is the core goal of solving equations. It means manipulating the equation so that the variable you're solving for is alone on one side of the equation. This process often involves several steps: using inverse operations such as addition/subtraction or multiplication/division, distributing, combining like terms, and factoring.

Remember that the operations should be performed in a particular order to ensure that you simplify correctly. It's similar to untangling a knot - you must be methodical to isolate the variable without complicating the equation further. This step-by-step approach will lead to a clear and simple expression of the variable.

Equation Checking

Once you believe you have solved the equation, it's vital to verify your solution by checking it. This process involves substituting the value you found for the variable back into the original equation to see if it satisfies the equation.

If both sides of the original equation remain equal after this substitution, the solution is correct. If they're not equal, then an error has occurred at some point in the solving process, requiring you to reevaluate your steps. This check acts as a quality assurance method, confirming that the isolation and manipulation of the variable were done correctly. It's an essential final step in solving linear equations and should not be overlooked.