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Answer :
Sure! Let's go through each part of the question step-by-step.
### 4. Explain why [tex]\(2(x-2)=50\)[/tex] is true for the same value of [tex]\(x\)[/tex] in the equation [tex]\(4(x-2)=100\)[/tex].
To begin, let's solve the first equation [tex]\(4(x-2) = 100\)[/tex] to find the value of [tex]\(x\)[/tex]:
1. Distribute the 4:
[tex]\(4(x - 2) = 4x - 8\)[/tex]
2. Set the equation equal to 100:
[tex]\(4x - 8 = 100\)[/tex]
3. Add 8 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\(4x = 108\)[/tex]
4. Divide both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\(x = \frac{108}{4}\)[/tex]
[tex]\(x = 27\)[/tex]
Now, let's check if [tex]\(2(x-2) = 50\)[/tex] is true for [tex]\(x = 27\)[/tex]:
1. Substitute [tex]\(x = 27\)[/tex] into the equation [tex]\(2(x-2)\)[/tex]:
[tex]\(2(27 - 2)\)[/tex]
2. Simplify inside the parentheses:
[tex]\(2(25)\)[/tex]
3. Multiply:
[tex]\(50\)[/tex]
This means that for [tex]\(x = 27\)[/tex], [tex]\(2(x-2) = 50\)[/tex] is also a true statement.
### 5. Solving the equation [tex]\(7.5d = 2.5d\)[/tex]
a. Will both moves lead to the solution?
- Lin's method was to divide both sides by [tex]\(2.5d\)[/tex]:
This approach assumes [tex]\(d \neq 0\)[/tex] and simplifies to [tex]\(3 = 1\)[/tex], which is incorrect. This method does not lead to a valid solution because we cannot divide by zero.
- Elena's method was to subtract [tex]\(2.5d\)[/tex] from each side:
[tex]\[
7.5d - 2.5d = 2.5d - 2.5d \\
5d = 0
\][/tex]
This results in [tex]\(d = 0\)[/tex], which is the correct solution.
b. What is the solution?
Given both methods, only Elena's approach correctly identifies the solution:
- The solution is: [tex]\(d = 0\)[/tex]
Therefore, the moves made by Lin won't lead to the correct solution, but Elena's method will yield the correct value of [tex]\(d\)[/tex].
### 4. Explain why [tex]\(2(x-2)=50\)[/tex] is true for the same value of [tex]\(x\)[/tex] in the equation [tex]\(4(x-2)=100\)[/tex].
To begin, let's solve the first equation [tex]\(4(x-2) = 100\)[/tex] to find the value of [tex]\(x\)[/tex]:
1. Distribute the 4:
[tex]\(4(x - 2) = 4x - 8\)[/tex]
2. Set the equation equal to 100:
[tex]\(4x - 8 = 100\)[/tex]
3. Add 8 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\(4x = 108\)[/tex]
4. Divide both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\(x = \frac{108}{4}\)[/tex]
[tex]\(x = 27\)[/tex]
Now, let's check if [tex]\(2(x-2) = 50\)[/tex] is true for [tex]\(x = 27\)[/tex]:
1. Substitute [tex]\(x = 27\)[/tex] into the equation [tex]\(2(x-2)\)[/tex]:
[tex]\(2(27 - 2)\)[/tex]
2. Simplify inside the parentheses:
[tex]\(2(25)\)[/tex]
3. Multiply:
[tex]\(50\)[/tex]
This means that for [tex]\(x = 27\)[/tex], [tex]\(2(x-2) = 50\)[/tex] is also a true statement.
### 5. Solving the equation [tex]\(7.5d = 2.5d\)[/tex]
a. Will both moves lead to the solution?
- Lin's method was to divide both sides by [tex]\(2.5d\)[/tex]:
This approach assumes [tex]\(d \neq 0\)[/tex] and simplifies to [tex]\(3 = 1\)[/tex], which is incorrect. This method does not lead to a valid solution because we cannot divide by zero.
- Elena's method was to subtract [tex]\(2.5d\)[/tex] from each side:
[tex]\[
7.5d - 2.5d = 2.5d - 2.5d \\
5d = 0
\][/tex]
This results in [tex]\(d = 0\)[/tex], which is the correct solution.
b. What is the solution?
Given both methods, only Elena's approach correctly identifies the solution:
- The solution is: [tex]\(d = 0\)[/tex]
Therefore, the moves made by Lin won't lead to the correct solution, but Elena's method will yield the correct value of [tex]\(d\)[/tex].
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