Problem 76 - Introductory Algebra for College Students

Question

Make Sense? Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I used the addition and multiplication properties of equality to solve \(3 x=20+4\)

Step-by-Step Solution

Answer

Yes, the statement 'I used the addition and multiplication properties of equality to solve \(3x=20+4\)' does make sense. The addition property of equality is used to simplify the equation by adding 20 to 4, and the inverse of the multiplication property, the division property of equality, is used to solve for 'x' correctly.

Step 1: Identify the equation

The equation given in the statement is \(3x = 20 + 4\). Here, 'x' is the unknown to be solved.

Step 2: Making use of addition

The first step is to add the constants on the right-hand side of the equation, which simplifies the equation to \(3x = 24\). This action demonstrates the use of the addition property of equality to simplify the equation.

Step 3: Solve for x using multiplication properties

Then, to solve for 'x', the equation can be divided by '3' on both sides. This simplifies the equation to \(x = 8\), demonstrating the use of the multiplication property of equality, although in a form of its inverse, the division property of equality.

Step 4: Verify the solution

By substituting the value of 'x' into the original equation, it is possible to confirm that \(3*8 = 20+4\), which simplifies to \(24 = 24\), confirming that 'x=8' is the solution. This verifies that the statement makes sense.

Key Concepts

Addition Property of Equality

The Addition Property of Equality is a fundamental aspect of solving linear equations. It allows us to add the same number to both sides of an equation without changing the equality. This is crucial because it maintains the balance of the equation. The idea is based on the notion that if two values are equal, then adding the same value to both of them keeps them equal.
In our exercise, consider the equation:

• Given: \(3x = 20 + 4\)
• We use the addition property by summing the constants on the right, \(20\) and \(4\), resulting in \(24\).

This simplification step is crucial as it makes the equation easier to solve. Instead of dealing with multiple terms, we now have a simpler form: \(3x = 24\). This sets the stage for solving the equation by the next steps using different properties.

Multiplication Property of Equality

The Multiplication Property of Equality is another essential tool in equation solving. It permits us to multiply both sides of an equation by the same non-zero number without affecting the equation's equality. In the context of our exercise, however, we deal more with the inverse of multiplication - division.
The goal is to isolate the variable \(x\). In the equation \(3x = 24\), \(x\) is being multiplied by 3. To solve for \(x\), we use the inverse operation of multiplication: division.

• Divide both sides by 3:
• \(\frac{3x}{3} = \frac{24}{3}\)

This simplification transforms the equation to \(x = 8\), indicating that \(x\) equals 8 is a solution of the equation. Thus, employing division, a form of this property, has enabled us to isolate the variable effectively.

Equation Simplification

Simplifying an equation is a process where we reduce it to its simplest form while maintaining equality. This process commonly involves combining like terms, applying basic arithmetic, and organizing the equation for easier handling. Simplification is critical because it prepares the equation for solving using various algebraic properties.
For example, in our original equation, \(3x = 20 + 4\), simplification involves adding the constants 20 and 4 to yield \(3x = 24\).
This step is often straightforward but essential, as it reduces potential errors in complex expressions. With a simplified equation, subsequent steps like applying the multiplication property of equality become significantly more manageable. Thus, simplification is woven into the entire problem-solving process, acting as a precursor to more complex operations.

Inverse Operations

Inverse operations are operations that undo each other, such as addition and subtraction, or multiplication and division. In the context of solving equations, inverse operations are central because they allow us to isolate variables.

• To isolate a variable, recognize what's being done to it, and then apply the opposite operation.

In the provided exercise, the variable \(x\) in \(3x = 24\) is being multiplied by 3. The inverse of multiplication is division, so we divide both sides by 3 to "undo" the multiplication:

• Let's see that in action: \(\frac{3x}{3} = \frac{24}{3}\)
• Simplifying gives us \(x = 8\).

Inverse operations help to untangle the equation and aid in pinpointing the value of the unknown. They are essentially the backbone of algebraic problem-solving, allowing us to systematically approach and solve equations.