A student is asked to find the solution to this equation:
[tex]$2 x-6=12$[/tex]
The student's work is shown:
[tex]$2 x-6=12+6=\frac{18}{2}=9$[/tex]
The student's equation is [ ] equation because it is [ ]. One way we know this is [ ] and 6 is [ ] to 9.
Answer (1)
The student's initial equation is analyzed, and the error in their steps is identified.
Correct algebraic manipulation is applied to both sides of the equation to isolate the variable x .
The correct solution x = 9 is derived by adding 6 to both sides and then dividing by 2.
The student's incorrect steps are explained, emphasizing the importance of maintaining equality on both sides of the equation. The student's equation is not a true equation because it is incorrect. One way we know this is that 12 and 6 is not equal to 9.
Explanation
Analyzing the Problem The student is trying to solve the equation 2 x − 6 = 12 . Let's analyze the student's work and identify the mistake.
Identifying the Error The student wrote: 2 x − 6 = 12 + 6 = 2 18 = 9 . This is where the error lies. The student seems to be trying to isolate x , but makes a mistake in the process. To correctly solve the equation, we need to isolate the term with x on one side of the equation.
Correcting the Equation To isolate 2 x , we should add 6 to both sides of the original equation: 2 x − 6 + 6 = 12 + 6 . This simplifies to 2 x = 18 .
Solving for x Now, to solve for x , we divide both sides of the equation 2 x = 18 by 2: 2 2 x = 2 18 , which gives x = 9 .
Explaining Why It's Not a True Equation The student's equation 2 x − 6 = 12 + 6 = 2 18 = 9 is not a true equation because the student incorrectly added 6 to only the right side in a flawed attempt to isolate x, and then divided by 2. The student should have added 6 to both sides of the original equation first, and then divided both sides by 2.
Verifying the Solution One way we know this is that 2 x − 6 = 12 , so if x = 9 , then 2 ( 9 ) − 6 = 18 − 6 = 12 , which is true. However, the student's steps are mathematically incorrect. Adding 6 to 12 and dividing by 2 is not a valid step in solving for x . The student is mixing up operations and not maintaining equality.
Final Answer Therefore, the student's equation is not a true equation because it is incorrect . One way we know this is that 12 and 6 is not equal to 9.
Examples
Imagine you're balancing a scale. The equation 2 x − 6 = 12 is like having something on one side of the scale ( 2 x − 6 ) that needs to equal something on the other side (12). To find the value of 'x', you need to perform the same operations on both sides to keep the scale balanced. If you only add or subtract from one side, the scale becomes unbalanced, and you won't find the correct value of 'x'. This principle is crucial in many real-life situations, such as managing budgets, calculating mixtures, or ensuring fairness in distributions.
