Answer :
To determine which equations have the same solution as the given equation [tex]\(2.3p - 10.1 = 6.5p - 4\)[/tex], we need to simplify and compare each option to see if it matches the original equation's solution.
First, let's simplify the original equation:
1. Start with the equation:
[tex]\(2.3p - 10.1 = 6.5p - 4\)[/tex].
2. Move the terms involving [tex]\(p\)[/tex] to one side by subtracting [tex]\(6.5p\)[/tex] from both sides:
[tex]\(2.3p - 6.5p = -4 + 10.1\)[/tex].
3. Simplify by combining like terms:
[tex]\(-4.2p = 6.1\)[/tex].
Now, let's compare the options:
1. Option 1: [tex]\(2.3p - 10.1 = 6.4p - 4\)[/tex]
- Simplify by moving terms involving [tex]\(p\)[/tex]:
[tex]\(2.3p - 6.4p = -4 + 10.1\)[/tex].
- Combine like terms:
[tex]\(-4.1p = 6.1\)[/tex].
- This does not match the original since [tex]\(-4.1p\)[/tex] is not the same as [tex]\(-4.2p\)[/tex].
2. Option 2: [tex]\(2.9 - 10.1 = 6.49p - 4\)[/tex]
- There is no consistent equation here that involves the variable [tex]\(p\)[/tex] in a similar way to compare directly.
3. Option 3: [tex]\(230p - 1010 = 650p - 400 - p\)[/tex]
- Simplify by moving terms involving [tex]\(p\)[/tex]:
[tex]\(230p - 650p + p = 400 - 1010\)[/tex].
- Combine like terms:
[tex]\(-419p = -610\)[/tex].
- Divide everything by 10:
[tex]\(-41.9p = -61\)[/tex].
- This matches in the form when simplified similarly as it results in an equivalent expression.
4. Option 4: [tex]\(23n - 101 - 65p - 10p\)[/tex]
- This includes an [tex]\(n\)[/tex] and does not correctly form a similar equation as required.
5. Option 5: [tex]\(2.3p - 14.1 = 6.4p - 4\)[/tex]
- Simplify by moving terms involving [tex]\(p\)[/tex]:
[tex]\(2.3p - 6.4p = -4 + 14.1\)[/tex].
- Combine like terms:
[tex]\(-4.1p = 10.1\)[/tex].
- This does not match the original since [tex]\(-4.1p\)[/tex] is not the same as [tex]\(-4.2p\)[/tex].
The correct option that results in the same type of equation as the original is:
- Option 3: [tex]\(230p - 1010 = 650p - 400 - p\)[/tex]
This option, when simplified, gives an equivalent relationship for the variable [tex]\(p\)[/tex] compared to the original expression.
First, let's simplify the original equation:
1. Start with the equation:
[tex]\(2.3p - 10.1 = 6.5p - 4\)[/tex].
2. Move the terms involving [tex]\(p\)[/tex] to one side by subtracting [tex]\(6.5p\)[/tex] from both sides:
[tex]\(2.3p - 6.5p = -4 + 10.1\)[/tex].
3. Simplify by combining like terms:
[tex]\(-4.2p = 6.1\)[/tex].
Now, let's compare the options:
1. Option 1: [tex]\(2.3p - 10.1 = 6.4p - 4\)[/tex]
- Simplify by moving terms involving [tex]\(p\)[/tex]:
[tex]\(2.3p - 6.4p = -4 + 10.1\)[/tex].
- Combine like terms:
[tex]\(-4.1p = 6.1\)[/tex].
- This does not match the original since [tex]\(-4.1p\)[/tex] is not the same as [tex]\(-4.2p\)[/tex].
2. Option 2: [tex]\(2.9 - 10.1 = 6.49p - 4\)[/tex]
- There is no consistent equation here that involves the variable [tex]\(p\)[/tex] in a similar way to compare directly.
3. Option 3: [tex]\(230p - 1010 = 650p - 400 - p\)[/tex]
- Simplify by moving terms involving [tex]\(p\)[/tex]:
[tex]\(230p - 650p + p = 400 - 1010\)[/tex].
- Combine like terms:
[tex]\(-419p = -610\)[/tex].
- Divide everything by 10:
[tex]\(-41.9p = -61\)[/tex].
- This matches in the form when simplified similarly as it results in an equivalent expression.
4. Option 4: [tex]\(23n - 101 - 65p - 10p\)[/tex]
- This includes an [tex]\(n\)[/tex] and does not correctly form a similar equation as required.
5. Option 5: [tex]\(2.3p - 14.1 = 6.4p - 4\)[/tex]
- Simplify by moving terms involving [tex]\(p\)[/tex]:
[tex]\(2.3p - 6.4p = -4 + 14.1\)[/tex].
- Combine like terms:
[tex]\(-4.1p = 10.1\)[/tex].
- This does not match the original since [tex]\(-4.1p\)[/tex] is not the same as [tex]\(-4.2p\)[/tex].
The correct option that results in the same type of equation as the original is:
- Option 3: [tex]\(230p - 1010 = 650p - 400 - p\)[/tex]
This option, when simplified, gives an equivalent relationship for the variable [tex]\(p\)[/tex] compared to the original expression.
