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Chapter 4: Problem 99

In Exercises 97-100, solve the equation and check your solution. $$ 4(5 x-6)-3(6 x+1)=0 $$

Short Answer

Expert verified
The solution to the equation is \(x = 13.5\).

Step by step solution

01

Distribute

First, distribute the 4 to the terms within the first set of parentheses and distribute -3 to the terms within the second set of parentheses. This results in: \(20x - 24 - 18x - 3 = 0\).
02

Combine like terms

Combine the x terms to simplify the equation: \(2x - 27 = 0\).
03

Isolate x

Next, isolate x by adding 27 to both sides of the equation. Now the equation is: \(2x = 27\). Then, divide both sides of the equation by 2 to solve for x: \(x = 13.5\).
04

Check

Check the solution by substituting \(x = 13.5\) back into the original equation: \(4(5 * 13.5 - 6) - 3(6 * 13.5 + 1)\). After simplification, both sides of the equation equal to 0, confirming that \(x = 13.5\) is the correct solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is an essential concept in solving linear equations, especially when dealing with expressions in parentheses. It allows us to multiply a single term outside the parentheses with each term inside the parentheses. In our example, we have the equation:
  • First part: \(4(5x - 6)\)
  • Second part: \(-3(6x + 1)\)
To apply the distributive property, multiply 4 by each term within the first parentheses, resulting in \(20x - 24\). Do the same for the second set, multiplying -3 by both terms, which gives \(-18x - 3\). This step transforms the equation, setting the stage for further simplification.
Combining Like Terms
After distributing, the next step is to combine like terms. Like terms have the same variable part, which can be constants or variable terms such as \(x\) terms. In the transformed equation:
  • \(20x - 18x\) are like terms because they both contain the variable \(x\).
  • When combined, they simplify to \(2x\).
  • Similarly, you combine the constants \(-24\) and \(-3\) to get \(-27\).
Therefore, the simplified equation is \(2x - 27 = 0\). This step reduces the complexity by consolidating similar components of the equation.
Isolating Variables
Isolating the variable is crucial for finding its value. The goal is to get the variable, in this case \(x\), on one side of the equation by itself. Starting from the simplified equation \(2x - 27 = 0\):
  • Add 27 to both sides to begin isolating \(x\).
  • This leads to \(2x = 27\).
  • Next, divide both sides by 2 to solve for \(x\), which results in \(x = 13.5\).
Thus, isolating the variable involves reversing operations such as subtraction or multiplication around \(x\) to determine its exact value.
Equation Verification
Once the variable is isolated and solved, it's crucial to verify the solution to ensure accuracy. Verification involves substituting the solved value of \(x\) back into the original equation to see if it holds true. For example, substitute \(x = 13.5\) back into:
  • \(4(5 * 13.5 - 6) - 3(6 * 13.5 + 1)\)
  • After following the order of operations, simplify the expression.
  • Check if both sides of the equation equate to 0.
When both sides are equal, it confirms that the solution is accurate. Verification proves the integrity of the mathematical process and the correctness of the solution.

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Most popular questions from this chapter

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