(120/s)+(480/2s)+(3x/s)==6

(150/s)+(570/2s)+3(x-60)/s =17/4

Let's solve this system of equations step by step. The given system is:

\[
\text{(1)} \quad \frac{120}{s} + \frac{480}{2s} + \frac{3x}{s} = 6
\]

\[
\text{(2)} \quad \frac{150}{s} + \frac{570}{2s} + \frac{3(x-60)}{s} = \frac{17}{4}
\]

### Step 1: Simplify both equations

**Equation (1):**

\[
\frac{120}{s} + \frac{480}{2s} + \frac{3x}{s} = 6
\]

Simplifying \(\frac{480}{2s}\) gives:

\[
\frac{120}{s} + \frac{240}{s} + \frac{3x}{s} = 6
\]

Combining terms:

\[
\frac{120 + 240 + 3x}{s} = 6
\]

\[
\frac{360 + 3x}{s} = 6
\]

Multiply both sides by \(s\):

\[
360 + 3x = 6s
\]

**Equation (1) becomes:**

\[
360 + 3x = 6s \quad \text{(Equation A)}
\]

---

**Equation (2):**

\[
\frac{150}{s} + \frac{570}{2s} + \frac{3(x-60)}{s} = \frac{17}{4}
\]

Simplifying \(\frac{570}{2s}\) gives:

\[
\frac{150}{s} + \frac{285}{s} + \frac{3(x-60)}{s} = \frac{17}{4}
\]

Expand \(3(x - 60)\):

\[
\frac{150}{s} + \frac{285}{s} + \frac{3x - 180}{s} = \frac{17}{4}
\]

Combine terms:

\[
\frac{150 + 285 + 3x - 180}{s} = \frac{17}{4}
\]

Simplifying:

\[
\frac{255 + 3x}{s} = \frac{17}{4}
\]

Multiply both sides by \(s\):

\[
255 + 3x = \frac{17}{4}s
\]

Multiply both sides by 4 to eliminate the fraction:

\[
4(255 + 3x) = 17s
\]

Simplify:

\[
1020 + 12x = 17s
\]

**Equation (2) becomes:**

\[
1020 + 12x = 17s \quad \text{(Equation B)}
\]

---

### Step 2: Solve the system

We now have two equations:

1. \( 360 + 3x = 6s \)
2. \( 1020 + 12x = 17s \)

#### Solving for \(s\) in terms of \(x\) from Equation (A):

From \(360 + 3x = 6s\), solve for \(s\):

\[
s = \frac{360 + 3x}{6}
\]

Substitute this expression for \(s\) in Equation (B):

\[
1020 + 12x = 17\left(\frac{360 + 3x}{6}\right)
\]

Simplify the right-hand side:

\[
1020 + 12x = \frac{17(360 + 3x)}{6}
\]

Multiply both sides by 6 to eliminate the denominator:

\[
6(1020 + 12x) = 17(360 + 3x)
\]

Simplify:

\[
6120 + 72x = 6120 + 51x
\]

Subtract 6120 from both sides:

\[
72x = 51x
\]

Subtract \(51x\) from both sides:

\[
21x = 0
\]

Solve for \(x\):

\[
x = 0
\]

### Step 3: Find \(s\)

Substitute \(x = 0\) back into Equation (A):

\[
360 + 3(0) = 6s
\]

\[
360 = 6s
\]

Solve for \(s\):

\[
s = \frac{360}{6} = 60
\]

### Final Solution:
\[
x = 0, \quad s = 60
\]

Would you like more details or explanations on any part?

---

Here are 5 related questions to explore further:
1. How do you solve systems of nonlinear equations in general?
2. What is the substitution method, and when is it useful?
3. What happens if the system has no solution or infinite solutions?
4. How can these types of equations be modeled in physics or economics?
5. How do the steps change when fractions or decimals are involved?

**Tip:** Always simplify your equations as much as possible before solving. This can save time and reduce errors.

(120/s)+(480/2s)+(3x/s)==6
(150/s)+(570/2s)+3(x-60)/s =17/4

Learn how to solve a system of rational equations involving variables s and x with a detailed step-by-step solution. The problem explores key algebraic concepts like simplifying rational expressions and solving systems using substitution, ideal for students in Grades 10-12.

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