Problem 22 - 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. $$-x=23$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -23\).

Step 1: Apply the Multiplication Property of Equality

To isolate 'x', multiply both sides of the equation by -1. This property of equality means if -x equals 23, then x equals -23.

Step 2: Substitute Value of 'x' to Check the Solution

Now we need to verify our solution for 'x'. Substitute -23 for 'x' in the original equation (-x = 23). The left side of the equation becomes --23 which is 23, and this matches with the other side of the equation, confirming that the solution is valid.

Key Concepts

Solving Algebraic Equations

Solving algebraic equations is a fundamental skill in mathematics which allows us to find the value of variables. An algebraic equation like \( -x = 23 \) is a statement of equality between two expressions, with a variable (in this case, 'x') that must be solved for. To solve for 'x', you apply various arithmetic operations, keeping in mind that whatever operation you do to one side of the equation, you must also do to the other. This ensures the balance of the equation is maintained.

Applying the Multiplication Property

When you come across an equation with a negative variable, like our example \( -x = 23 \), you can apply the multiplication property of equality. This involves multiplying (or dividing) both sides of the equation by the same nonzero number to isolate the variable. For this equation, multiplying by -1 yields \( x = -23 \) as -1*(-x) is 'x', and -1*23 is -23. It's crucial to perform the same operation on both sides to maintain the equality.

Equation Properties

Properties of Equality

Equations operate on the principle of balance. This adherence to the balance is safeguarded by certain properties, one of them being the multiplication property of equality applied in our exercise. The property states that if two expressions are equal, they remain equal if multiplied by the same nonzero value. This property ensures consistency in the equation's balance and is instrumental in solving algebraic equations

Other fundamental properties include the addition, subtraction, and division properties of equality, all of which allow for the manipulation of equations to isolate variables. Each property has a specific use depending on the form of the equation and the desired method of isolating the variable.

Verifying Solutions in Algebra

After finding the potential solution to an algebraic equation, it's vital to verify that the solution is indeed correct. This process includes substituting the found value back into the original equation to see if it satisfies the equation's balance. In the given exercise, \( x = -23 \) is substituted back into \( -x = 23 \) to give \( --23 \), which simplifies to \( 23 \), thus confirming the equality.

Verification serves as an error-check mechanism and is a good practice to ensure the dismissal of extraneous solutions that might have resulted from the algebraic manipulations. In more complex equations, especially those with multiple variables or solutions, this step is particularly important.