Question

select the correct answer.
what is the solution for ( r ) in the equation?
( 0.4r - 0.13 - 0.7r = 3r - 5.74 )
( \bigcirc r = 2.1 )
( \bigcirc r = 1.7 )
( \bigcirc r = -1.7 )
( \bigcirc r = -2.1 )

Explanation:

Step1: Combine like terms on the left side

Combine the \( r \)-terms: \( 0.4r - 0.7r = -0.3r \), so the equation becomes \( -0.3r - 0.13 = 3r - 5.74 \).

Step2: Move all \( r \)-terms to one side

Add \( 0.3r \) to both sides: \( -0.13 = 3r + 0.3r - 5.74 \), which simplifies to \( -0.13 = 3.3r - 5.74 \).

Step3: Move constant terms to the other side

Add \( 5.74 \) to both sides: \( -0.13 + 5.74 = 3.3r \), so \( 5.61 = 3.3r \).

Step4: Solve for \( r \)

Divide both sides by \( 3.3 \): \( r = \frac{5.61}{3.3} = 1.7 \). Wait, no, wait, let's check again. Wait, in step 2, when we add \( 0.3r \) to both sides, it's \( -0.13 = 3r + 0.3r - 5.74 \), so \( 3r + 0.3r = 3.3r \), then add \( 5.74 \) to both sides: \( -0.13 + 5.74 = 3.3r \), \( 5.61 = 3.3r \), then \( r = 5.61 / 3.3 = 1.7 \)? Wait, but let's check the original equation. Wait, no, maybe I made a sign error. Let's start over.

Original equation: \( 0.4r - 0.13 - 0.7r = 3r - 5.74 \)

Combine left \( r \)-terms: \( 0.4r - 0.7r = -0.3r \), so \( -0.3r - 0.13 = 3r - 5.74 \)

Now, subtract \( 3r \) from both sides: \( -0.3r - 3r - 0.13 = -5.74 \)

So \( -3.3r - 0.13 = -5.74 \)

Now, add \( 0.13 \) to both sides: \( -3.3r = -5.74 + 0.13 \)

\( -3.3r = -5.61 \)

Now, divide both sides by \( -3.3 \): \( r = (-5.61) / (-3.3) = 1.7 \)? Wait, no, \( -5.61 / -3.3 = 1.7 \)? Wait, \( 3.3 \times 1.7 = 5.61 \), yes. Wait, but let's check the options. Wait, the options include \( r = 1.7 \), but let's check with \( r = 1.7 \):

Left side: \( 0.4(1.7) - 0.13 - 0.7(1.7) = 0.68 - 0.13 - 1.19 = 0.68 - 1.32 = -0.64 \)

Right side: \( 3(1.7) - 5.74 = 5.1 - 5.74 = -0.64 \). Oh, so it works. Wait, but earlier I thought maybe a sign error, but no, it's correct. Wait, but let's check the steps again.

Wait, original equation: \( 0.4r - 0.13 - 0.7r = 3r - 5.74 \)

Left: \( (0.4r - 0.7r) - 0.13 = -0.3r - 0.13 \)

Right: \( 3r - 5.74 \)

Now, bring all \( r \) to left and constants to right:

\( -0.3r - 3r = -5.74 + 0.13 \)

\( -3.3r = -5.61 \)

Then \( r = (-5.61)/(-3.3) = 1.7 \). Yes, that's correct. So the solution is \( r = 1.7 \).

Answer:

\( r = 1.7 \) (corresponding to the option "r = 1.7")