Karen found that the solution to [tex]x-7+5 x=36[/tex] is [tex]x=6[/tex]. Which of these could be the way she found the solution?
A. add -7 and [tex]5 x[/tex], subtract [tex]x[/tex] from both sides of the equation
B. add [tex]x+5 x[/tex], subtract 7 from both sides of the equation
C. add [tex]x-7+5 x[/tex], add 36 to both sides of the equation
D. add [tex]x+5 x[/tex], add 7 to both sides of the equation
Answer (2)
Combine like terms in the equation: x − 7 + 5 x = 36 becomes 6 x − 7 = 36 .
Add 7 to both sides of the equation: 6 x = 43 .
Divide both sides by 6 to solve for x : x = 6 43 .
The correct method is to add x + 5 x and add 7 to both sides, leading to the solution 6 43 .
Explanation
Analyze the equation First, let's analyze the given equation: x − 7 + 5 x = 36 . Karen found the solution to be x = 6 . However, if we substitute x = 6 into the equation, we get 6 − 7 + 5 ( 6 ) = 6 − 7 + 30 = 29 , which is not equal to 36. So, x = 6 is NOT the correct solution. The correct solution can be found as follows:
Combine the like terms on the left side of the equation: x + 5 x − 7 = 6 x − 7 . So, the equation becomes 6 x − 7 = 36 .
Examine the options Now, let's examine the given options to see which one describes a valid step in solving the equation.
Option 1: add -7 and 5 x , subtract x from both sides of the equation. This doesn't directly lead to isolating x. It's not the most efficient way, but let's see. Adding -7 and 5 x doesn't change the equation. Subtracting x from both sides would give − 7 + 5 x = 36 − x , which isn't helpful.
Option 2: add x + 5 x , subtract 7 from both sides of the equation. Adding x + 5 x results in 6 x − 7 = 36 . Subtracting 7 from both sides gives 6 x − 14 = 29 , which is not helpful.
Option 3: add x − 7 + 5 x , add 36 to both sides of the equation. This is not a valid step because adding the entire left side to both sides doesn't simplify the equation or isolate x. Adding 36 to both sides gives x − 7 + 5 x + 36 = 36 + 36 , which simplifies to 6 x + 29 = 72 , not helpful.
Option 4: add x + 5 x , add 7 to both sides of the equation. Adding x + 5 x gives 6 x − 7 = 36 . Adding 7 to both sides gives 6 x = 36 + 7 , which simplifies to 6 x = 43 . This is a valid step towards isolating x.
Solve the equation The correct steps to solve the equation x − 7 + 5 x = 36 are:
Combine like terms: 6 x − 7 = 36
Add 7 to both sides: 6 x = 43
Divide by 6: x = 6 43
Conclusion Therefore, the way Karen could have found the solution is by adding x + 5 x and then adding 7 to both sides of the equation.
Examples
When solving for an unknown variable in an equation, like determining the cost of an item after tax, we use similar algebraic steps. For instance, if an item costs x dollars, and a 7% tax is added, resulting in a total cost of 36 , w ec an w r i t e t h ee q u a t i o n x + 0.07x = 36$. Combining like terms gives 1.07 x = 36 . Dividing both sides by 1.07 isolates x , giving the original cost of the item before tax. This principle of isolating variables is fundamental in various real-world calculations.
The correct method for solving the equation x − 7 + 5 x = 36 involves first combining like terms to get 6 x − 7 = 36 , then isolating x. The option that describes this method is B , where x and 5x are added, and then 7 is subtracted from both sides. However, the value of x mentioned as 6 is not correct when checked against the original equation.
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