Question

linear equations and inequalities
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.
2.3p - 10.1 = 6.49p - 4
2.3p - 10.1 = 6.4p - 4
2.3p - 14.1 = 6.4p - 4
230p - 1010 = 650p - 400 - p

Explanation:

Step1: Simplify the right side of the original equation

The original equation is \(2.3p - 10.1=6.5p - 4 - 0.01p\). Combine like terms on the right side: \(6.5p-0.01p = 6.49p\), so the equation becomes \(2.3p - 10.1 = 6.49p - 4\). Let's check each option:

Step2: Analyze Option 1 (\(2.3p - 10.1 = 6.49p - 4\))

We just simplified the original equation to this form, so this equation has the same solution as the original.

Step3: Analyze Option 2 (\(2.3p - 10.1 = 6.4p - 4\))

The right side of the original equation after combining like terms is \(6.49p-4\), not \(6.4p - 4\), so this is incorrect.

Step4: Analyze Option 3 (\(2.3p - 14.1 = 6.4p - 4\))

The left side of the original equation is \(2.3p - 10.1\), not \(2.3p - 14.1\), and the right side is also incorrect, so this is wrong.

Step5: Analyze Option 4 (\(230p - 1010 = 650p - 400 - p\))

Multiply both sides of the original equation \(2.3p - 10.1 = 6.5p - 4 - 0.01p\) by 100 to eliminate decimals.
Left side: \(100\times(2.3p - 10.1)=230p - 1010\)
Right side: \(100\times(6.5p - 4 - 0.01p)=650p - 400 - p\) (since \(100\times6.5p = 650p\), \(100\times(- 4)=-400\), \(100\times(-0.01p)=-p\))
So this equation is equivalent to the original equation (multiplied both sides by 100, which is a valid operation that doesn't change the solution), so it has the same solution.

Answer:

A. \(2.3p - 10.1 = 6.49p - 4\)
D. \(230p - 1010 = 650p - 400 - p\)