Select the correct answer.
Which value of [tex]$x$[/tex] makes the equation true?
[tex]$\frac{3 x-6}{3}=\frac{7 x-3}{6}$[/tex]
A. -11
B. -9
C. -3
D. 3
Answer (2)
Multiply both sides of the equation by 6 to eliminate the fractions: 2 ( 3 x − 6 ) = 7 x − 3 .
Distribute the 2 on the left side: 6 x − 12 = 7 x − 3 .
Subtract 6 x from both sides: − 12 = x − 3 .
Add 3 to both sides to solve for x : x = − 9 .
Explanation
Understanding the Problem We are given the equation 3 3 x − 6 = 6 7 x − 3 and we need to find the value of x that makes this equation true.
Eliminating Fractions To solve for x , we first eliminate the fractions by multiplying both sides of the equation by 6 (the least common multiple of 3 and 6): 6 ⋅ 3 3 x − 6 = 6 ⋅ 6 7 x − 3
Simplifying the Equation This simplifies to: 2 ( 3 x − 6 ) = 7 x − 3
Distributing Next, we distribute the 2 on the left side of the equation: 6 x − 12 = 7 x − 3
Isolating x Now, we want to isolate x . We can subtract 6 x from both sides of the equation: − 12 = x − 3
Solving for x Finally, we add 3 to both sides of the equation to solve for x :
− 12 + 3 = x − 9 = x
Final Answer So, the value of x that makes the equation true is x = − 9 .
Examples
In real life, solving linear equations like this can help determine the break-even point for a business. For example, if the cost of producing x items is represented by 6 7 x − 3 and the revenue from selling x items is represented by 3 3 x − 6 , finding the value of x where these two expressions are equal tells you how many items you need to sell to cover your costs. This type of problem also appears in physics, when calculating motion with constant acceleration or in chemistry, when dealing with concentrations in solutions.
The solution to the equation 3 3 x − 6 = 6 7 x − 3 is x = − 9 . Therefore, the correct answer is option B. -9.
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