Problem 78 - Introductory Algebra for College Students

Question

Make Sense? Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When I use the addition and multiplication properties to solve \(2 x+5=17,\) I undo the operations in the opposite order in which they are performed.

Step-by-Step Solution

Answer

The statement makes sense. The operations are indeed undone in the reverse order in which they were originally performed when solving this type of equation.

Step 1: Understanding the Statement

The statement says that operations (multiplication and addition in this case) are undone in the reverse order when using addition and multiplication properties to solve equations. This means that if the equation is \(2x + 5 = 17\), allowing x to be the subject of the formula, the addition operation should be 'undone' first, before the multiplication operation.

Step 2: Assessing the Statement

To undo the operations in the equation, whatever is added or subtracted is done so before any multiplication or division when the equation is being solved. This means that for the equation \(2x + 5 = 17\), subtracting 5 from both sides will be done before dividing both sides by 2.

Step 3: Evaluating the Statement

This means that the statement is correct. The addition operation (which is adding 5) is undone before the multiplication operation (which is multiplying by 2).

Key Concepts

Addition Property of Equality

The Addition Property of Equality is a fundamental concept used to solve equations. This property states that when you add or subtract the same amount from both sides of an equation, the equality of the equation is maintained. For example, in the equation \(2x + 5 = 17\), the addition of 5 can be removed by subtracting 5 on both sides.

• Initial Equation: \(2x + 5 = 17\)
• Subtract 5 from both sides: \(2x + 5 - 5 = 17 - 5\)
• Simplified Equation: \(2x = 12\)

This step is crucial because it begins to isolate the variable, making it easier to solve the equation. Think of it as balancing scales; whatever you do to one side must be done to the other to keep them equal.
This principle helps in gradually simplifying complex equations, ensuring that the equality remains intact throughout the process.

Multiplication Property of Equality

The Multiplication Property of Equality allows you to solve equations by multiplying or dividing both sides by the same non-zero number. It is used after the addition or subtraction operations have been addressed, especially when dealing with coefficients attached to variables.
Taking our equation from before, after applying the Addition Property of Equality, we have \(2x = 12\). To completely solve for \(x\), we employ the Multiplication Property of Equality.

• Equation: \(2x = 12\)
• Divide both sides by 2: \(\frac{2x}{2} = \frac{12}{2}\)
• Simplified Solution: \(x = 6\)

In this step, we 'undo' the multiplication by 2 by dividing, effectively isolating the variable \(x\). This property ensures that operations like multiplication and division do not affect the equality of the original equation, maintaining the solution's accuracy.

Order of Operations

The Order of Operations is a set of rules that determines the order in which operations are carried out when simplifying or solving equations. A common abbreviation used to remember this order is PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
In solving equations, especially when reversing operations as seen in the equation \(2x + 5 = 17\), it's important to follow the reverse of this order. This means addressing addition or subtraction first, followed by multiplication or division.

• First, undo addition/subtraction: subtract 5 to get \(2x = 12\).
• Next, undo multiplication/division: divide by 2 to find \(x = 6\).

Following this order ensures that you properly isolate the variable while maintaining the integrity of the equality throughout your solution. It avoids mistakes that could arise from mixing operations or reversing steps inadvertently.