https://file.aiquickdraw.com/mathbot/file/1730487268290epc0p3ie.jpg

Let's solve each system of equations step-by-step.

---

### Problem (a)
The system is:
\[
\begin{cases} 
2x - 4y = 10 \\
x + 5y = 40 
\end{cases}
\]

1. **Solve for \( x \) from the second equation**:
   \[
   x + 5y = 40 \implies x = 40 - 5y
   \]

2. **Substitute \( x = 40 - 5y \) into the first equation**:
   \[
   2(40 - 5y) - 4y = 10
   \]
   Expanding and simplifying:
   \[
   80 - 10y - 4y = 10
   \]
   \[
   80 - 14y = 10
   \]
   \[
   -14y = -70
   \]
   \[
   y = 5
   \]

3. **Substitute \( y = 5 \) back into \( x = 40 - 5y \)**:
   \[
   x = 40 - 5(5) = 40 - 25 = 15
   \]

**Solution for (a):** \( (x, y) = (15, 5) \)

---

### Problem (b)
The system is:
\[
\begin{cases} 
3x - 5y = 4 \\
-2x + 6y = 18 
\end{cases}
\]

1. **Multiply the first equation by 2 and the second equation by 3 to align the coefficients of \( x \)**:
   \[
   \begin{cases} 
   6x - 10y = 8 \\
   -6x + 18y = 54 
   \end{cases}
   \]

2. **Add the equations to eliminate \( x \)**:
   \[
   (6x - 10y) + (-6x + 18y) = 8 + 54
   \]
   \[
   8y = 62
   \]
   \[
   y = \frac{62}{8} = \frac{31}{4}
   \]

3. **Substitute \( y = \frac{31}{4} \) back into the first equation to find \( x \)**:
   \[
   3x - 5\left(\frac{31}{4}\right) = 4
   \]
   \[
   3x - \frac{155}{4} = 4
   \]
   \[
   3x = 4 + \frac{155}{4} = \frac{16 + 155}{4} = \frac{171}{4}
   \]
   \[
   x = \frac{171}{4} \div 3 = \frac{171}{12} = \frac{57}{4}
   \]

**Solution for (b):** \( \left( x, y \right) = \left( \frac{57}{4}, \frac{31}{4} \right) \)

---

### Final Answers:
- (a) \( (15, 5) \)
- (b) \( \left( \frac{57}{4}, \frac{31}{4} \right) \)

Would you like more details on any part of these solutions?

---

Here are some additional questions to explore:
1. How do you know when to add or subtract equations when solving a system?
2. What are other methods for solving systems of equations (besides substitution and elimination)?
3. How can you verify the solution of a system of equations?
4. What happens if a system of equations has no solutions?
5. How can graphing help understand solutions to systems of equations?

**Tip:** When solving a system, check if you can simplify first by aligning coefficients for easier elimination.

Solve each system of equations. a) { 2x - 4y = 10, x + 5y = 40 } b) { 3x - 5y = 4, -2x + 6y = 18 }

Learn how to solve systems of equations with two variables using substitution and elimination. This example includes step-by-step solutions for systems like {2x - 4y = 10, x + 5y = 40} and {3x - 5y = 4, -2x + 6y = 18}. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation