Problem 28 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve 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: Distribute and Simplify

Distribute the 4 in the second part of the equation. The equation becomes \(100=-(x-1) + 4x - 24\). Then, simplify the equation by eliminating the brackets, the equation now becomes \(100= -x + 1 + 4x - 24\)

Step 2: Collect Like Terms

In the equation \(100= -x + 1 + 4x - 24\), \( -x\) and \( 4x\) are like terms and can be added together. Doing that, the equation becomes \(100= 3x -23\).

Step 3: Solve for x

Add 23 to both sides of the equation to isolate the term with x on one side: \(100 + 23 = 3x - 23 + 23\). Simplified, the equation becomes \(123= 3x\). Then divide both sides by 3 to solve for x: \(\frac{123}{3} = \frac{3x}{3}\). Simplified, the solution is \(x = 41\)

Step 4: Check the Solution

Substitute 41 for x in the original equation: \(100=-(41-1)+4(41-6)\). Performing the operations, the equation becomes \(100= -40 +140\). This simplifies to \(100=100\). The left hand side equals the right hand side, hence the solution is verified.

Key Concepts

Distributive Property

When solving linear equations, the distributive property is a fundamental technique that helps simplify expressions. It involves multiplying a single term by each term inside a parenthesis or bracket. Let's consider the equation from the exercise: \(100=-(x-1)+4(x-6)\). Here, the distributive property is applied to the term \(4(x-6)\).

• Multiply 4 by \(x\): gives \(4x\)
• Multiply 4 by -6: gives \(-24\)

Thus, \(4(x-6)\) expands to \(4x - 24\). Likewise, when dealing with negative signs, remember that they distribute similarly. For instance, the term \(-1(x-1)\) becomes \(-x + 1\). Understanding and using the distributive property appropriately makes the rest of the solving process significantly easier.

Combining Like Terms

After using the distributive property, combining like terms is crucial in further simplifying the equation. Like terms are terms that have the same variable raised to the same power. For our example, once we have the expression \(-x + 1 + 4x - 24\), we can combine terms with \(x\) and constant terms separately.

• Combine \(-x\) and \(4x\). Since they both contain the variable \(x\), they simplify to \(3x\).
• Combine the constants 1 and -24. Combining gives us -23.

Through combining like terms, the expression is further simplified to \(3x - 23\). This step is all about grouping and simplifying, which helps you isolate the variable more effectively.

Equation Verification

Equation verification is the final and important step in solving equations. It ensures that the proposed solution satisfies the original equation. In our example, finding \(x = 41\) as our solution means we need to check if this value works in the starting equation: \(100=-(x-1)+4(x-6)\). By substituting \(x\) with 41, you perform calculations:

• Firstly, replace \(x\) with 41: \(-(41-1) + 4(41-6)\)
• Simplify: \(-40 + 140\)
• The equation becomes \(100 = 100\).

Since both sides of the equation are equal, we confirm that \(x = 41\) is indeed the correct solution. Verification is key in confirming accuracy and ensuring no errors occurred during solving.