What is the solution for $x$ in the equation?
$-3 x+7 x-8=34+9 x-2$
A. $x=-\frac{1}{8}$
B. $x=8$
C. $x=\frac{1}{8}$
D. $x=-8$
Answer (2)
Combine like terms on both sides of the equation: 4 x − 8 = 32 + 9 x .
Isolate the variable terms on one side and the constants on the other: − 40 = 5 x .
Solve for x by dividing both sides by 5: x = − 8 .
The solution is − 8 .
Explanation
Problem Analysis We are given the equation − 3 x + 7 x − 8 = 34 + 9 x − 2 . Our goal is to solve for x .
Simplifying the Equation First, we simplify both sides of the equation by combining like terms. On the left side, we have − 3 x + 7 x = 4 x . So the left side becomes 4 x − 8 . On the right side, we have 34 − 2 = 32 . So the right side becomes 32 + 9 x . Now our equation is 4 x − 8 = 32 + 9 x .
Isolating the Variable Next, we want to isolate the x terms on one side and the constant terms on the other side. We can subtract 4 x from both sides to get − 8 = 32 + 5 x . Then, we subtract 32 from both sides to get − 8 − 32 = 5 x , which simplifies to − 40 = 5 x .
Solving for x Now, we solve for x by dividing both sides by 5 : 5 − 40 = x , which gives us x = − 8 .
Checking the Solution Finally, we check our solution by substituting x = − 8 back into the original equation: − 3 ( − 8 ) + 7 ( − 8 ) − 8 = 34 + 9 ( − 8 ) − 2 . This simplifies to 24 − 56 − 8 = 34 − 72 − 2 , which further simplifies to − 40 = − 40 . Since the equation holds true, our solution is correct.
Final Answer Therefore, the solution for x is − 8 .
Examples
Solving linear equations is a fundamental skill in algebra and has numerous real-world applications. For instance, imagine you are managing a budget for a project. You have a fixed amount of money and need to allocate it to different tasks. If each task has a cost that depends on a variable (like time or resources), you can set up a linear equation to determine the optimal value of that variable to stay within your budget. For example, if you have a budget of $1000 and each hour of work costs $50, you can determine how many hours you can afford to work by solving the equation 50 x = 1000 , where x is the number of hours. This ensures you don't overspend and can complete the project successfully.
The solution to the equation − 3 x + 7 x − 8 = 34 + 9 x − 2 is x = − 8 . This was achieved by simplifying both sides of the equation, isolating the variable, and verifying the solution. The chosen multiple-choice option is D. x = − 8 .
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