What is the value of $x$ in the equation $2.5(6 x-4)=10+4(1.5+0.5 x)$?
Answer (2)
Distribute the constants on both sides of the equation: 15 x − 10 = 10 + 6 + 2 x .
Combine like terms: 15 x − 10 = 16 + 2 x .
Isolate the x term: 13 x = 26 .
Solve for x : x = 2 .
Explanation
Problem Setup We are given the equation 2.5 ( 6 x − 4 ) = 10 + 4 ( 1.5 + 0.5 x ) and we want to find the value of x .
Distributing Constants First, distribute the constants on both sides of the equation: 2.5 ( 6 x ) − 2.5 ( 4 ) = 10 + 4 ( 1.5 ) + 4 ( 0.5 x ) 15 x − 10 = 10 + 6 + 2 x 15 x − 10 = 16 + 2 x
Isolating x Terms Next, we want to isolate the x terms on one side of the equation. Subtract 2 x from both sides: 15 x − 2 x − 10 = 16 + 2 x − 2 x 13 x − 10 = 16
Isolating the x Term Now, add 10 to both sides of the equation to isolate the x term: 13 x − 10 + 10 = 16 + 10 13 x = 26
Solving for x Finally, divide both sides by 13 to solve for x :
13 13 x = 13 26 x = 2
Final Answer Therefore, the value of x is 2.
Examples
In real-world scenarios, solving linear equations like this can help determine the break-even point in business. For example, if x represents the number of units you need to sell to cover your costs, this equation could model the relationship between revenue and expenses. By solving for x , you find the number of units needed to make revenue equal expenses, which is a crucial step in business planning and financial analysis. Understanding how to manipulate and solve such equations is fundamental in making informed business decisions.
The value of x in the equation 2.5 ( 6 x − 4 ) = 10 + 4 ( 1.5 + 0.5 x ) is 2. This was found by distributing constants, isolating the x terms, and solving for x . Understanding how to manipulate and solve linear equations is essential in many real-world applications.
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