Solve and check the following linear equation:
[tex]7 x-(3 x-8)=44[/tex]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is [tex] \{9\} [/tex].
(Type an integer or a simplified fraction.)
B. The solution set is [tex] \{x \mid x [/tex] is all real number[tex] \} [/tex].
C. The solution set is [tex] \varnothing [/tex].
Answer (2)
Distribute the negative sign: 7 x − 3 x + 8 = 44 .
Combine like terms: 4 x + 8 = 44 .
Subtract 8 from both sides: 4 x = 36 .
Divide by 4 to solve for x: x = 9 . The solution set is { 9 } .
Explanation
Understanding the Problem We are given the equation 7 x − ( 3 x − 8 ) = 44 . Our goal is to solve for x , meaning we want to find the value of x that makes this equation true.
Simplifying the Equation First, we need to simplify the left side of the equation. We distribute the negative sign in front of the parentheses: 7 x − 3 x + 8 = 44 .
Combining Like Terms Next, we combine the like terms on the left side of the equation. We have 7 x − 3 x , which simplifies to 4 x . So the equation becomes 4 x + 8 = 44 .
Isolating the Variable Term Now, we want to isolate the term with x on one side of the equation. To do this, we subtract 8 from both sides of the equation: 4 x + 8 − 8 = 44 − 8 , which simplifies to 4 x = 36 .
Solving for x Finally, we want to solve for x . To do this, we divide both sides of the equation by 4: 4 4 x = 4 36 , which simplifies to x = 9 .
Checking the Solution Now, we need to check our solution to make sure it is correct. We substitute x = 9 back into the original equation: 7 ( 9 ) − ( 3 ( 9 ) − 8 ) = 44 .
Verifying the Solution We simplify the expression: 63 − ( 27 − 8 ) = 44 . Then, 63 − 19 = 44 , which simplifies to 44 = 44 . Since the equation holds true, our solution is correct.
Final Answer Therefore, the solution set is { 9 } .
Examples
Linear equations are used in various real-life scenarios, such as calculating the cost of items. For example, suppose you want to buy several items at a store. If each item costs a fixed amount, and you have a coupon that reduces the total cost, you can use a linear equation to determine how many items you can buy with a certain budget. Let's say each item costs $7, and you have a coupon for $8 off your entire purchase. If you have $44 to spend, the equation 7x - 8 = 44 can help you find out how many items (x) you can purchase.
The solution to the equation 7 x − ( 3 x − 8 ) = 44 is x = 9 . After checking by substituting back into the original equation, everything holds true. Therefore, the answer is oxed{\{9\}} .
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