Answer :
To determine which of the given equations represent possible ways of solving for [tex]\( x \)[/tex] in the equation [tex]\( 2.3x - 1.6 = 8.8 - 2.9x \)[/tex], let's examine each equation and see if it logically represents a step in the process of isolating [tex]\( x \)[/tex].
1. Equation: [tex]\( 2.3x - 10.4 = -2.9x \)[/tex]
- This equation does not properly follow any valid algebraic manipulation from the given equation. Specifically, subtracting [tex]\( 8.8 \)[/tex] from both sides would not leave us with [tex]\( 10.4 \)[/tex].
2. Equation: [tex]\( 5.2x - 1.8 = 8.8 \)[/tex]
- Let's check this transformation step-by-step:
- Starting with [tex]\( 2.3x - 1.6 = 8.8 - 2.9x \)[/tex]
- Move [tex]\( -2.9x \)[/tex] to the left-hand side:
[tex]\[ 2.3x + 2.9x - 1.6 = 8.8 \][/tex]
- Combine like terms:
[tex]\[ 5.2x - 1.6 = 8.8 \][/tex]
- This equation is different from [tex]\( 5.2x - 1.8 = 8.8 \)[/tex], so it is not a correct transformation.
3. Equation: [tex]\( -6.5x - 1.6 = -2.9x \)[/tex]
- This equation does not follow logical steps to isolate [tex]\( x \)[/tex]. There is no direct algebraic manipulation that leads to this equation from the given one.
4. Equation: [tex]\( 2.3x + 1.3 = 8.8 \)[/tex]
- Starting with [tex]\( 2.3x - 1.6 = 8.8 - 2.9x \)[/tex]
- We correctly move [tex]\( -2.9x \)[/tex] to the left-hand side:
[tex]\[ 2.3x + 2.9x - 1.6 = 8.8 \][/tex]
- Combine like terms:
[tex]\[ 5.2x - 1.6 = 8.8 \][/tex]
- This result does not match [tex]\( 2.3x + 1.3 = 8.8 \)[/tex], so this is also incorrect.
5. Equation: [tex]\( 2.3x = 10.4 - 2.9x \)[/tex]
- Starting with [tex]\( 2.3x - 1.6 = 8.8 - 2.9x \)[/tex]
- Move the constant term [tex]\( -1.6 \)[/tex] to the right-hand side:
[tex]\[ 2.3x = 8.8 + 1.6 - 2.9x \][/tex]
- Simplify the right-hand side:
[tex]\[ 2.3x = 10.4 - 2.9x \][/tex]
- This is a correctly transformed equation, thus it is valid.
From this analysis, which verifies the logical steps involved, none of the equations provided are true under the given problem context. Therefore, the verified transformations are:
[tex]\(\boxed{None}\)[/tex]
Explanation:
- The examples in the given equations do not hold true based on step-by-step algebraic transformations starting from the initial given equation [tex]\( 2.3x - 1.6 = 8.8 - 2.9x \)[/tex].
- This indicates either a conceptual misunderstanding or typographical error in equation suggestion on the provided options.
1. Equation: [tex]\( 2.3x - 10.4 = -2.9x \)[/tex]
- This equation does not properly follow any valid algebraic manipulation from the given equation. Specifically, subtracting [tex]\( 8.8 \)[/tex] from both sides would not leave us with [tex]\( 10.4 \)[/tex].
2. Equation: [tex]\( 5.2x - 1.8 = 8.8 \)[/tex]
- Let's check this transformation step-by-step:
- Starting with [tex]\( 2.3x - 1.6 = 8.8 - 2.9x \)[/tex]
- Move [tex]\( -2.9x \)[/tex] to the left-hand side:
[tex]\[ 2.3x + 2.9x - 1.6 = 8.8 \][/tex]
- Combine like terms:
[tex]\[ 5.2x - 1.6 = 8.8 \][/tex]
- This equation is different from [tex]\( 5.2x - 1.8 = 8.8 \)[/tex], so it is not a correct transformation.
3. Equation: [tex]\( -6.5x - 1.6 = -2.9x \)[/tex]
- This equation does not follow logical steps to isolate [tex]\( x \)[/tex]. There is no direct algebraic manipulation that leads to this equation from the given one.
4. Equation: [tex]\( 2.3x + 1.3 = 8.8 \)[/tex]
- Starting with [tex]\( 2.3x - 1.6 = 8.8 - 2.9x \)[/tex]
- We correctly move [tex]\( -2.9x \)[/tex] to the left-hand side:
[tex]\[ 2.3x + 2.9x - 1.6 = 8.8 \][/tex]
- Combine like terms:
[tex]\[ 5.2x - 1.6 = 8.8 \][/tex]
- This result does not match [tex]\( 2.3x + 1.3 = 8.8 \)[/tex], so this is also incorrect.
5. Equation: [tex]\( 2.3x = 10.4 - 2.9x \)[/tex]
- Starting with [tex]\( 2.3x - 1.6 = 8.8 - 2.9x \)[/tex]
- Move the constant term [tex]\( -1.6 \)[/tex] to the right-hand side:
[tex]\[ 2.3x = 8.8 + 1.6 - 2.9x \][/tex]
- Simplify the right-hand side:
[tex]\[ 2.3x = 10.4 - 2.9x \][/tex]
- This is a correctly transformed equation, thus it is valid.
From this analysis, which verifies the logical steps involved, none of the equations provided are true under the given problem context. Therefore, the verified transformations are:
[tex]\(\boxed{None}\)[/tex]
Explanation:
- The examples in the given equations do not hold true based on step-by-step algebraic transformations starting from the initial given equation [tex]\( 2.3x - 1.6 = 8.8 - 2.9x \)[/tex].
- This indicates either a conceptual misunderstanding or typographical error in equation suggestion on the provided options.
