Problem 28 - Introductory Algebra for College Students

Question

Solve each equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. $$100=-(x-1)+4(x-6)$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 41 \)

Step 1: Expand the equation.

Applying the distributive property, the given equation becomes: \(100 = -x + 1 + 4x - 24 \).

Step 2: Combine like terms.

Next, combine the like terms from previous step to simplify the equation: \(100 = -x + 4x + 1 - 24 \), which simplifies to \(100 = 3x - 23\).

Step 3: Rearrange the equation to isolate variable x.

Now, add 23 to both sides of the equation and then divide by 3 to isolate the variable \(x \). That gives: \(x = (100 + 23)/3\).

Step 4: Compute the value of x.

Perform the operations to compute the value of \( x \). Find \( x = 123/3 = 41.\)

Step 5: Verify the solution.

Now plug in the value of \( x = 41 \) into the original equation. The left-hand side equals the right-hand side, which confirms that the solution is correct.

Key Concepts

Algebraic Problem Solving

Algebraic problem solving is a critical aspect of mathematics that deals with finding unknown variables. In essence, it's like being a detective in the world of numbers and operations. To solve an algebraic problem, particularly a linear equation like the one in our exercise, you follow a series of steps that simplify the problem until the unknown variable is isolated.

It starts with understanding and identifying the type of equation you're dealing with. From there, you apply various algebraic operations, such as distribution, combining like terms, and isolating the variable. Each step brings you closer to revealing the value of the unknown variable. It's important to work carefully through each step to avoid small errors that could lead to incorrect solutions.

• Understand the problem
• Identify the type of equation
• Apply algebraic operations
• Isolate the variable
• Check the solution

Practicing algebraic problem solving is essential, as it enhances logical thinking and problem-solving skills that are valuable not just in math, but in real-world situations as well.

Distributive Property

The distributive property is a cornerstone of algebra that allows you to multiply a single term by each term within a parenthesis. It's essentially a way of spreading out or distributing the multiplication.

The general form of the distributive property is expressed as: \( a(b + c) = ab + ac \). In our exercise, the distributive property was used when \( -(x - 1) + 4(x - 6) \) was expanded into \( -x + 1 + 4x - 24 \).

Why use the distributive property?

It simplifies equations and allows for further steps, such as combining like terms, to be executed with ease. Employing this property correctly is crucial for simplifying expressions and solving equations efficiently.

Combining Like Terms

Combining like terms is simplifying an algebraic expression by merging terms that have the same variables raised to the same power. Think of it as grouping together similar items. For example, if you have 2 apples and you get 3 more apples, you now have 5 apples. In algebra, if you have \( 2x \) and you add \( 3x \), you have \( 5x \).

In the exercise, after applying the distributive property, we combined the like terms \( -x \) and \( 4x \) to simplify the equation to \( 3x - 23 \). This step is crucial because it pares down the equation to its most basic form, making it easier to find the unknown variable.

Equation Verification

Equation verification is the final step in solving algebraic equations. It’s like proofreading in writing or quality control in manufacturing. In this phase, you ensure that your solution is correct by plugging the value back into the original equation. It's the seal of approval for your solution.

For our problem, we verify the solution \( x = 41 \) by substituting it back into the original equation, confirming that both sides of the equation balance out. This step prevents any oversight that might have occurred during the solving process. Always remember, even when you're confident about your solution, a quick check can save you from unnoticed errors.