Answer :
We begin with the equation
[tex]$$
2.3p - 10.1 = 6.5p - 4 - 0.01p.
$$[/tex]
Step 1. Simplify the Right Side
Notice that the right-hand side contains like terms in [tex]\( p \)[/tex]. Combine the terms:
[tex]$$
6.5p - 0.01p = 6.49p.
$$[/tex]
Thus the equation becomes:
[tex]$$
2.3p - 10.1 = 6.49p - 4.
$$[/tex]
This is exactly the equation given in Option 2:
[tex]$$
2.3p - 10.1 = 6.49p - 4.
$$[/tex]
Step 2. Multiply Through to Eliminate Decimals
To remove the decimals, multiply both sides of the original simplified equation by 100:
[tex]\[
\begin{aligned}
100(2.3p - 10.1) &= 100(6.49p - 4) \\
230p - 1010 &= 649p - 400.
\end{aligned}
\][/tex]
This new equation matches Option 3, which is written as
[tex]$$
230p - 1010 = 650p - 400 - p.
$$[/tex]
Notice on the right-hand side:
[tex]$$
650p - p = 649p,
$$[/tex]
so the equation becomes
[tex]$$
230p - 1010 = 649p - 400.
$$[/tex]
Thus, Option 3 is equivalent to the original equation.
Step 3. Verify the Other Options
- Option 1: The equation is
[tex]$$
2.3p - 10.1 = 6.4p - 4.
$$[/tex]
Here the coefficient of [tex]\( p \)[/tex] on the right is [tex]\( 6.4 \)[/tex], which does not match [tex]\( 6.49 \)[/tex] in the simplified equation.
- Option 4: The equation is
[tex]$$
23p - 101 = 65p - 40 - p.
$$[/tex]
Simplify the right-hand side:
[tex]$$
65p - p = 64p.
$$[/tex]
Thus the equation becomes
[tex]$$
23p - 101 = 64p - 40.
$$[/tex]
Multiplying the simplified original equation by 10, we obtain
[tex]$$
23p - 101 = 64.9p - 40,
$$[/tex]
which does not match the right-hand side of Option 4.
- Option 5: This option is not a valid rewriting of the equation.
Final Answer
The equations that have the same solution as the original equation are Option 2 and Option 3.
[tex]$$
2.3p - 10.1 = 6.5p - 4 - 0.01p.
$$[/tex]
Step 1. Simplify the Right Side
Notice that the right-hand side contains like terms in [tex]\( p \)[/tex]. Combine the terms:
[tex]$$
6.5p - 0.01p = 6.49p.
$$[/tex]
Thus the equation becomes:
[tex]$$
2.3p - 10.1 = 6.49p - 4.
$$[/tex]
This is exactly the equation given in Option 2:
[tex]$$
2.3p - 10.1 = 6.49p - 4.
$$[/tex]
Step 2. Multiply Through to Eliminate Decimals
To remove the decimals, multiply both sides of the original simplified equation by 100:
[tex]\[
\begin{aligned}
100(2.3p - 10.1) &= 100(6.49p - 4) \\
230p - 1010 &= 649p - 400.
\end{aligned}
\][/tex]
This new equation matches Option 3, which is written as
[tex]$$
230p - 1010 = 650p - 400 - p.
$$[/tex]
Notice on the right-hand side:
[tex]$$
650p - p = 649p,
$$[/tex]
so the equation becomes
[tex]$$
230p - 1010 = 649p - 400.
$$[/tex]
Thus, Option 3 is equivalent to the original equation.
Step 3. Verify the Other Options
- Option 1: The equation is
[tex]$$
2.3p - 10.1 = 6.4p - 4.
$$[/tex]
Here the coefficient of [tex]\( p \)[/tex] on the right is [tex]\( 6.4 \)[/tex], which does not match [tex]\( 6.49 \)[/tex] in the simplified equation.
- Option 4: The equation is
[tex]$$
23p - 101 = 65p - 40 - p.
$$[/tex]
Simplify the right-hand side:
[tex]$$
65p - p = 64p.
$$[/tex]
Thus the equation becomes
[tex]$$
23p - 101 = 64p - 40.
$$[/tex]
Multiplying the simplified original equation by 10, we obtain
[tex]$$
23p - 101 = 64.9p - 40,
$$[/tex]
which does not match the right-hand side of Option 4.
- Option 5: This option is not a valid rewriting of the equation.
Final Answer
The equations that have the same solution as the original equation are Option 2 and Option 3.
