Question

use properties to rewrite the given equation. which equations have the same solution as 2.3p - 10.1 = 6.5p - 4 - 0.01p? choose two correct answers. 230p - 1010 = 650p - 400 - p; 2.3p - 14.1 = 6.4p - 4; 2.3p - 10.1 = 6.49p - 4; 2.3p - 10.1 = 6.4p - 4; 23p - 101 = 65p - 40 - p

Explanation:

Step1: Simplify the original equation

Original equation: \(2.3p - 10.1 = 6.5p - 4 - 0.01p\)
Combine like terms on the right: \(6.5p - 0.01p=6.49p\), so equation becomes \(2.3p - 10.1 = 6.49p - 4\)

Step2: Analyze each option

- Option 1: \(230p - 1010 = 650p - 400 - p\)
Multiply original equation by 100: \(230p - 1010 = 650p - 400 - p\) (since \(2.3p\times100 = 230p\), \(10.1\times100 = 1010\), \(6.5p\times100 = 650p\), \(4\times100 = 400\), \(0.01p\times100 = p\)). But wait, original right side after combining is \(6.49p\), here it's \(650p - p=649p\), which is \(6.49p\times100\), so actually this is scaling by 100, but let's check others.
- Option 2: \(2.3p - 14.1 = 6.4p - 4\)
Left side: \(2.3p - 10.1 - 4=2.3p - 14.1\), right side: \(6.5p - 0.01p=6.49p eq6.4p\). Incorrect.
- Option 3: \(2.3p - 10.1 = 6.49p - 4\)
Matches the simplified original equation (from Step1). Correct.
- Option 4: \(2.3p - 10.1 = 6.4p - 4\)
Right side: \(6.5p - 0.01p = 6.49p eq6.4p\). Incorrect.
- Option 5: \(23p - 101 = 65p - 40 - p\)
Multiply original equation by 10: \(23p - 101 = 65p - 40 - p\) ( \(2.3p\times10 = 23p\), \(10.1\times10 = 101\), \(6.5p\times10 = 65p\), \(4\times10 = 40\), \(0.01p\times10 = 0.1p\)? Wait no, wait original right side is \(6.5p - 4 - 0.01p\), multiplying by 10: \(65p - 40 - 0.1p\), but here it's \(65p - 40 - p\). Wait, no—wait original equation after combining is \(2.3p - 10.1 = 6.49p - 4\). Multiply by 10: \(23p - 101 = 64.9p - 40\), but this option has \(65p - p=64p\), which is wrong. Wait, earlier mistake: in Option 1, \(650p - p = 649p\), which is \(6.49p\times100\), so that's correct scaling. Wait, no, let's recalculate Option 5:
Original equation: \(2.3p - 10.1 = 6.5p - 4 - 0.01p\)
Multiply both sides by 10: \(23p - 101 = 65p - 40 - 0.1p\). But Option 5 is \(23p - 101 = 65p - 40 - p\) (which is \(65p - p = 64p\)), which is incorrect. Wait, I made a mistake earlier. Let's re-express:

Wait, the correct simplification from original:
Original: \(2.3p - 10.1 = 6.5p - 4 - 0.01p\)
Combine right: \(6.5p - 0.01p = 6.49p\), so \(2.3p - 10.1 = 6.49p - 4\) (Option 3).

Now, Option 1: Multiply original equation by 100:
Left: \(2.3p\times100 - 10.1\times100 = 230p - 1010\)
Right: \(6.5p\times100 - 4\times100 - 0.01p\times100 = 650p - 400 - p\)
Which is \(230p - 1010 = 649p - 400\) (since \(650p - p = 649p\)), and original after multiplying by 100 is \(230p - 1010 = 649p - 400\), which is same as Option 1. Wait, so Option 1 is also correct? Wait no, original right side after combining is \(6.49p - 4\), multiplying by 100 gives \(649p - 400\), and Option 1 is \(650p - 400 - p = 649p - 400\), so yes, Option 1 is correct. Wait, but earlier when I checked Option 3, it's \(2.3p - 10.1 = 6.49p - 4\), which is the original equation simplified. So two correct answers: Option 1 (\(230p - 1010 = 650p - 400 - p\)) and Option 3 (\(2.3p - 10.1 = 6.49p - 4\))? Wait no, let's check Option 5 again. Wait, maybe I messed up Option 1's right side. Wait \(6.5p - 0.01p = 6.49p\), so original equation is \(2.3p - 10.1 = 6.49p - 4\).

Option 1: Multiply by 100: \(230p - 1010 = 650p - 400 - p\) (since \(6.5p\times100 = 650p\), \(0.01p\times100 = p\), so \(650p - p = 649p\), and \(6.49p\times100 = 649p\); \(4\times100 = 400\), \(10.1\times100 = 1010\)). So yes, Option 1 is equivalent (multiplying by 100).

Option 3: \(2.3p - 10.1 = 6.49p - 4\) (combining like terms on right, which is what we did in Step1).

Wait, but the options given: let's list them again:

1. \(230p - 1010 = 650p - 400 - p\)
2. \(2.3p - 14.1 = 6.4p - 4\)
3. \(2.3p - 10.1 = 6.49p - 4\)
4. \(2.3p - 10.1 = 6.4p - 4\)
5. \(23p - 101 = 65p - 40 - p\)

So correct ones are Option 1 (scaling by 100) and Option 3 (combining like terms). Wait, but let's verify with solving for \(p\):

Original equation: \(2.3p - 10.1 = 6.5p - 4 - 0.01p\)
\(2.3p - 10.1 = 6.49p - 4\)
\(-10.1 + 4 = 6.49p - 2.3p\)
\(-6.1 = 4.19p\)
\(p = -6.1 / 4.19 \approx -1.456\)

Option 1: \(230p - 1010 = 650p - 400 - p\)
\(230p - 1010 = 649p - 400\)
\(-1010 + 400 = 649p - 230p\)
\(-610 = 419p\)
\(p = -610 / 419 \approx -1.456\) (same as original, since we multiplied by 100, solution scales? Wait no, actually, when you multiply both sides by a non - zero number, the solution remains the same. Wait, \(-6.1/4.19=-610/419\) (multiply numerator and denominator by 100), so yes, same solution.

Option 3: \(2.3p - 10.1 = 6.49p - 4\)
Same as original simplified, so solution same.

So the two correct answers are the first option (\(230p - 1010 = 650p - 400 - p\)) and the third option (\(2.3p - 10.1 = 6.49p - 4\)). Wait, but let's check Option 5: \(23p - 101 = 65p - 40 - p\)
\(23p - 101 = 64p - 40\)
\(-101 + 40 = 64p - 23p\)
\(-61 = 41p\)
\(p = -61/41\approx -1.487\), which is different. So Option 5 is wrong.

So correct options are:
1. \(230p - 1010 = 650p - 400 - p\)
3. \(2.3p - 10.1 = 6.49p - 4\)

Answer:

1. \(230p - 1010 = 650p - 400 - p\)
2. \(2.3p - 10.1 = 6.49p - 4\)