Sure! Let's find out which of these equations have the same value for [tex]\( x \)[/tex] as the given equation [tex]\(\frac{3}{5}(30x - 15) = 72\)[/tex].
1. Solving the Original Equation:
[tex]\[
\frac{3}{5}(30x - 15) = 72
\][/tex]
Step 1: Eliminate the fraction by multiplying both sides by 5:
[tex]\[
3(30x - 15) = 360
\][/tex]
Step 2: Simplify inside the parentheses:
[tex]\[
90x - 45 = 360
\][/tex]
Step 3: Add 45 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
90x = 405
\][/tex]
Step 4: Divide both sides by 90 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{405}{90} = 4.5
\][/tex]
So, the value of [tex]\( x \)[/tex] is [tex]\( 4.5 \)[/tex].
2. Checking Given Equations for the Same Value of [tex]\( x \)[/tex]:
(1) Equation: [tex]\( 18x - 15 = 72 \)[/tex]
Substitute [tex]\( x = 4.5 \)[/tex]:
[tex]\[
18(4.5) - 15 = 81 - 15 = 66 \quad \text{(does not equal 72)}
\][/tex]
This equation does not have the same value of [tex]\( x \)[/tex].
(2) Equation: [tex]\( 50x - 25 = 72 \)[/tex]
Substitute [tex]\( x = 4.5 \)[/tex]:
[tex]\[
50(4.5) - 25 = 225 - 25 = 200 \quad \text{(does not equal 72)}
\][/tex]
This equation does not have the same value of [tex]\( x \)[/tex].
(3) Equation: [tex]\( 18x - 9 = 72 \)[/tex]
Substitute [tex]\( x = 4.5 \)[/tex]:
[tex]\[
18(4.5) - 9 = 81 - 9 = 72
\][/tex]
This equation has the same value of [tex]\( x \)[/tex].
(4) Equation: [tex]\( 3(6x - 3) = 72 \)[/tex]
Substitute [tex]\( x = 4.5 \)[/tex]:
[tex]\[
3(6(4.5) - 3) = 3(27 - 3) = 3(24) = 72
\][/tex]
This equation has the same value of [tex]\( x \)[/tex].
(5) Equation: [tex]\( x = 4.5 \)[/tex]
This directly matches the value [tex]\( x = 4.5 \)[/tex].
Therefore, the three equations that have the same value for [tex]\( x \)[/tex] as [tex]\(\frac{3}{5}(30x - 15) = 72\)[/tex] are:
1. [tex]\( 18x - 9 = 72 \)[/tex]
2. [tex]\( 3(6x - 3) = 72 \)[/tex]
3. [tex]\( x = 4.5 \)[/tex]
Hence, the correct options are:
- [tex]\( 18x - 9 = 72 \)[/tex]
- [tex]\( 3(6x - 3) = 72 \)[/tex]
- [tex]\( x = 4.5 \)[/tex]
