Let's determine the equation that represents the point at which the cost of both ski slopes is the same.
The problem is asking us to find which equation out of the given four options balances out to the same cost for both ski slopes. To solve this, I'll guide you through each given option and compare it to the values we obtained.
First, let’s examine each equation provided in the options:
1. Option 1: [tex]\( 15x - 75 = 10x - 50 \)[/tex]
Subtract [tex]\(10x\)[/tex] from both sides:
[tex]\[
15x - 10x - 75 = -50
\][/tex]
Simplifying further:
[tex]\[
5x - 75 = -50
\][/tex]
Adding 75 to both sides:
[tex]\[
5x = 25
\][/tex]
Dividing both sides by 5:
[tex]\[
x = 5
\][/tex]
2. Option 2: [tex]\( 15x - 50 = 10x - 75 \)[/tex]
Subtract [tex]\(10x\)[/tex] from both sides:
[tex]\[
15x - 10x - 50 = -75
\][/tex]
Simplifying further:
[tex]\[
5x - 50 = -75
\][/tex]
Adding 50 to both sides:
[tex]\[
5x = -25
\][/tex]
Dividing both sides by 5:
[tex]\[
x = -5
\][/tex]
3. Option 3: [tex]\( 15x + 50 = 10x + 75 \)[/tex]
Subtract [tex]\(10x\)[/tex] from both sides:
[tex]\[
15x - 10x + 50 = 75
\][/tex]
Simplifying further:
[tex]\[
5x + 50 = 75
\][/tex]
Subtracting 50 from both sides:
[tex]\[
5x = 25
\][/tex]
Dividing both sides by 5:
[tex]\[
x = 5
\][/tex]
4. Option 4: [tex]\( 15x + 75 = 10x + 50 \)[/tex]
Subtract [tex]\(10x\)[/tex] from both sides:
[tex]\[
15x - 10x + 75 = 50
\][/tex]
Simplifying further:
[tex]\[
5x + 75 = 50
\][/tex]
Subtracting 75 from both sides:
[tex]\[
5x = -25
\][/tex]
Dividing both sides by 5:
[tex]\[
x = -5
\][/tex]
Now let's check which result makes the described costs equal.
From analysis, we see that [tex]\((15x - 75) - (10x - 50) = -60\)[/tex] gives us the equation.
Thus, Option 1: [tex]\(15x - 75=10x - 50\)[/tex] is the correct answer where both ski slopes have the same cost under the conditions given.
