Problem 78 - Introductory and Intermediate Algebra for College Students 4th

Question

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 does make sense. When solving equations, the operations are undone in the reverse order in which they were initially performed, which is the principle this statement adheres to.

Step 1 - Understanding Order of Operations

The statement brings up the concept of order of operations, which refers to the sequence in which mathematical operations are performed to simplify an equation or numerical expression. In generally accepted mathematic principles, the order of operations is Parentheses, Exponents, Multiplication and Division from left to right, and Addition and Subtraction from left to right, known as the acronym PEMDAS. But when solving equations, the operations are performed in reverse order of PEMDAS.

Step 2 - Breaking Down The Equation Solving Process

The given equation is \(2x + 5 = 17\). Using order of operations, the multiplication (2x) was performed before the addition (+5). However, when undoing the operations to isolate \(x\), the operations need to be performed in reverser order. That means, undoing the addition before undoing the multiplication. So, subtract 5 from both sides of the equation first to get \(2x = 12\), then divide both sides by 2 to solve for \(x = 6\).

Step 3 - Evaluating The Statement

Given the aforementioned process, it can be seen that indeed the addition and multiplication properties were used to solve the equation \(2x + 5 = 17\), and this was done by undoing the operations in the opposite order in which they were performed. Hence, upon analysis, the statement 'makes sense'.

Key Concepts

Order of Operations

When working with equations, it's crucial to remember the order of operations. This is a standard process that helps everyone understand mathematics the same way. The order of operations tells us the steps to follow when solving any math problem and is often remembered by the acronym PEMDAS.

• Parentheses
• Exponents
• Multiplication & Division (from left to right)
• Addition & Subtraction (from left to right)

But here's a twist! When you're solving equations, you'll actually perform these steps in the reverse order of PEMDAS. This might sound confusing, but think of it this way: when you solve equations, you need to "undo" operations to find the value of the unknown variable. By doing this in reverse, you isolate the variable correctly.

Addition Property

The addition property in algebra is like a trusty tool in your math toolkit. It says if you add the same number to both sides of an equation, the equation still holds true. This property helps keep the equation balanced.
To demonstrate, if you have a simple equation like \( x + 3 = 7 \), you can subtract 3 from both sides to keep it balanced. So you'll have \( x + 3 - 3 = 7 - 3 \), which simplifies to \( x = 4 \).
In the original equation \( 2x + 5 = 17 \), the addition property was used when we performed the reverse operation: subtracting 5 from both sides to help isolate the term with the variable, leading to \( 2x = 12 \). By using the addition property in this way, we're one step closer to finding the solution for \( x \).

Multiplication Property

The multiplication property of equations is another handy tool. It ensures that multiplying or dividing both sides of an equation by the same non-zero number keeps the equation balanced.
Imagine you have the equation \( x/2 = 3 \). To solve this, you would multiply both sides by 2 to isolate and solve for \( x \). It becomes \( x = 6 \). This property is all about maintaining balance while removing coefficients or fractions in front of a variable.
In the example equation \( 2x + 5 = 17 \), after using the addition property to simplify it to \( 2x = 12 \), we used the multiplication property by dividing both sides by 2. This operation provided the solution \( x = 6 \), giving us the final answer. The key is remembering that multiplication (and division) operations should be executed after handling addition and subtraction when working backwards through an equation.