Math Problem Statement

Follow the​ Step-by-Step process to solve the linear equation. ​a) Determine the LCD of all denominators in the given linear equation. ​b) Write the new linear equation obtained after multiplying both sides of the original equation by the LCD. ​c) Solve the equation. StartFraction x plus 6 Over 3 EndFraction minus StartFraction x minus 18 Over 4 EndFraction equals 3 Question content area bottom Part 1 ​a) The LCD is    12. ​(Simplify your​ answer.) Part 2 ​b) What is the new​ equation? A. 4​(xplus6​)minus3​(xminus18​)equals36 B. 12​(xplus6​)minus12​(xminus18​)equals36 C. 4​(xplus6​)minus3​(xminus18​)equals3 D. ​(xplus6​)minus​(xminus18​)equals36 Part 3 ​c) What is the​ solution? Select the correct choice​ and, if​ necessary, fill in the answer box to complete your choice. A. xequals    enter your response here ​(Type an integer or a simplified​ fraction.) B. The solution is all real numbers. C.

Solution

Let's solve the linear equation step by step.

Given Equation:

x+63x184=3\frac{x + 6}{3} - \frac{x - 18}{4} = 3

Part 1: Determine the LCD

The denominators in the equation are 3 and 4. The Least Common Denominator (LCD) of 3 and 4 is 12.

Part 2: Write the New Equation

Now, we'll multiply every term in the equation by the LCD (12) to eliminate the fractions:

12×(x+63)12×(x184)=12×312 \times \left(\frac{x + 6}{3}\right) - 12 \times \left(\frac{x - 18}{4}\right) = 12 \times 3

Simplifying each term:

4(x+6)3(x18)=364(x + 6) - 3(x - 18) = 36

So, the correct new equation is: 4(x+6)3(x18)=36\boxed{4(x + 6) - 3(x - 18) = 36}

This corresponds to option A.

Part 3: Solve the Equation

Now, let's solve the equation 4(x+6)3(x18)=364(x + 6) - 3(x - 18) = 36.

  1. Distribute the 4 and -3: 4x+243x+54=364x + 24 - 3x + 54 = 36

  2. Combine like terms: (4x3x)+(24+54)=36(4x - 3x) + (24 + 54) = 36 x+78=36x + 78 = 36

  3. Subtract 78 from both sides to isolate xx: x=3678x = 36 - 78 x=42x = -42

Thus, the solution is: x=42\boxed{x = -42}

This corresponds to option A with x=42x = -42.


Do you want more details or have any questions? Here are 5 related questions that could help deepen your understanding:

  1. What would the equation look like if the LCD was incorrectly calculated as 6 instead of 12?
  2. How do you check if x=42x = -42 is the correct solution by substituting it back into the original equation?
  3. Why is it important to find the LCD when dealing with equations involving fractions?
  4. What happens if you multiply the equation by a number other than the LCD?
  5. How would the process change if the original equation had more complex fractions?

Tip: Always simplify fractions first, if possible, to make the process of finding the LCD easier.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Fractions
Least Common Denominator (LCD)

Formulas

Multiplying both sides of an equation by the LCD to eliminate fractions

Theorems

Properties of Equality
Distributive Property

Suitable Grade Level

Grades 8-10