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Answer :
Sure! Let's break down and explain each part of the problem in more detail:
1. Equation [tex]\(x+6 = x+1\)[/tex]:
- To solve for [tex]\(x\)[/tex], subtract [tex]\(x\)[/tex] from both sides:
[tex]\[ 6 = 1 \][/tex]
- This statement is false, which means there's no value of [tex]\(x\)[/tex] that can satisfy the original equation. Hence, there is no solution.
2. Analyzing Noah's work:
- Original equation: [tex]\(2(5+x)-1=3x+9\)[/tex]
- Apply the distributive property:
[tex]\[ 10 + 2x - 1 = 3x + 9 \][/tex]
[tex]\[ 2x + 9 = 3x + 9 \][/tex]
- Subtract 10 from each side, which is a correct move:
[tex]\[ 2x - 1 = 3x - 1 \][/tex]
- Add 1 to each side, another correct move:
[tex]\[ 2x = 3x \][/tex]
- Subtracting [tex]\(2x\)[/tex] from both sides gives:
[tex]\[ 0 = x \][/tex]
- If we solve for [tex]\(x\)[/tex], we would try to isolate [tex]\(x\)[/tex], but instead it results in a contradiction: [tex]\(2 = 3\)[/tex]. Thus, the process ends with a false statement indicating no solution.
3. Lesson Synthesis - Analyzing Equations:
- A. [tex]\(\begin{array}{rl} 5(x-3) &= 5 \\ x-3 &= 1 \end{array}\)[/tex]
- Divide both sides by 5 correctly. The solutions for both forms are equivalent.
- B. [tex]\(\begin{aligned} 5z-3 &= 5 \\ 5z &= 2 \end{aligned}\)[/tex]
- Subtract 3 from each side. The change is valid and both equations can be resolved similarly.
- C. [tex]\(\begin{array}{c} 5(x-3) = 5x \\ x-3 = z \end{array}\)[/tex]
- The move does not maintain equality, rendering it incorrect since it introduces a variable without a math operation justifying it.
- D. [tex]\(\begin{array}{c} (5-3)x = 5z \\ 5-3 = 5 \end{array}\)[/tex]
- Simplifying incorrectly since 5-3 is not equal to 5, making this transformation invalid.
4. Cool Down Questions:
- 1. For [tex]\(4(x-2)=100\)[/tex], solving for [tex]\(x\)[/tex] gives [tex]\(x = 27\)[/tex]. Thus, when you scale both sides fairly, [tex]\(2(x-2)=50\)[/tex] is true because it represents the half of the former equation. Both point to the same value for [tex]\(x\)[/tex].
- 2.
- a. Dividing [tex]\(7.5d = 2.5d\)[/tex] by [tex]\(2.5d\)[/tex] does not help solve the equation because it incorrectly simplifies to a false statement (3 = 1). Subtracting [tex]\(2.5d\)[/tex] from each side simplifies to:
[tex]\[ 7.5d - 2.5d = 0 \][/tex]
Which reduces to [tex]\(5d = 0\)[/tex], leading us to the solution.
- b. Solution is [tex]\(d = 0\)[/tex]. This is the correct solution from properly working through the arithmetic steps.
1. Equation [tex]\(x+6 = x+1\)[/tex]:
- To solve for [tex]\(x\)[/tex], subtract [tex]\(x\)[/tex] from both sides:
[tex]\[ 6 = 1 \][/tex]
- This statement is false, which means there's no value of [tex]\(x\)[/tex] that can satisfy the original equation. Hence, there is no solution.
2. Analyzing Noah's work:
- Original equation: [tex]\(2(5+x)-1=3x+9\)[/tex]
- Apply the distributive property:
[tex]\[ 10 + 2x - 1 = 3x + 9 \][/tex]
[tex]\[ 2x + 9 = 3x + 9 \][/tex]
- Subtract 10 from each side, which is a correct move:
[tex]\[ 2x - 1 = 3x - 1 \][/tex]
- Add 1 to each side, another correct move:
[tex]\[ 2x = 3x \][/tex]
- Subtracting [tex]\(2x\)[/tex] from both sides gives:
[tex]\[ 0 = x \][/tex]
- If we solve for [tex]\(x\)[/tex], we would try to isolate [tex]\(x\)[/tex], but instead it results in a contradiction: [tex]\(2 = 3\)[/tex]. Thus, the process ends with a false statement indicating no solution.
3. Lesson Synthesis - Analyzing Equations:
- A. [tex]\(\begin{array}{rl} 5(x-3) &= 5 \\ x-3 &= 1 \end{array}\)[/tex]
- Divide both sides by 5 correctly. The solutions for both forms are equivalent.
- B. [tex]\(\begin{aligned} 5z-3 &= 5 \\ 5z &= 2 \end{aligned}\)[/tex]
- Subtract 3 from each side. The change is valid and both equations can be resolved similarly.
- C. [tex]\(\begin{array}{c} 5(x-3) = 5x \\ x-3 = z \end{array}\)[/tex]
- The move does not maintain equality, rendering it incorrect since it introduces a variable without a math operation justifying it.
- D. [tex]\(\begin{array}{c} (5-3)x = 5z \\ 5-3 = 5 \end{array}\)[/tex]
- Simplifying incorrectly since 5-3 is not equal to 5, making this transformation invalid.
4. Cool Down Questions:
- 1. For [tex]\(4(x-2)=100\)[/tex], solving for [tex]\(x\)[/tex] gives [tex]\(x = 27\)[/tex]. Thus, when you scale both sides fairly, [tex]\(2(x-2)=50\)[/tex] is true because it represents the half of the former equation. Both point to the same value for [tex]\(x\)[/tex].
- 2.
- a. Dividing [tex]\(7.5d = 2.5d\)[/tex] by [tex]\(2.5d\)[/tex] does not help solve the equation because it incorrectly simplifies to a false statement (3 = 1). Subtracting [tex]\(2.5d\)[/tex] from each side simplifies to:
[tex]\[ 7.5d - 2.5d = 0 \][/tex]
Which reduces to [tex]\(5d = 0\)[/tex], leading us to the solution.
- b. Solution is [tex]\(d = 0\)[/tex]. This is the correct solution from properly working through the arithmetic steps.
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