https://r2.erweima.ai/mathbot/image/17266497908448qzvbb51.jpg

The image contains several systems of linear equations in two variables. Let's solve them step by step:

### a)
\[
3x + y = 18
\]
\[
2x - y = 7
\]

1. Add both equations to eliminate \(y\):
\[
(3x + y) + (2x - y) = 18 + 7
\]
\[
5x = 25 \quad \Rightarrow \quad x = 5
\]

2. Substitute \(x = 5\) into the first equation:
\[
3(5) + y = 18 \quad \Rightarrow \quad 15 + y = 18 \quad \Rightarrow \quad y = 3
\]

Solution: \(x = 5, y = 3\)

---

### b)
\[
4x + 3y = 2
\]
\[
5x - 3y = 16
\]

1. Add both equations to eliminate \(y\):
\[
(4x + 3y) + (5x - 3y) = 2 + 16
\]
\[
9x = 18 \quad \Rightarrow \quad x = 2
\]

2. Substitute \(x = 2\) into the first equation:
\[
4(2) + 3y = 2 \quad \Rightarrow \quad 8 + 3y = 2 \quad \Rightarrow \quad 3y = -6 \quad \Rightarrow \quad y = -2
\]

Solution: \(x = 2, y = -2\)

---

### c)
\[
12x - 5y = 36
\]
\[
2x + 5y = 6
\]

1. Add both equations to eliminate \(y\):
\[
(12x - 5y) + (2x + 5y) = 36 + 6
\]
\[
14x = 42 \quad \Rightarrow \quad x = 3
\]

2. Substitute \(x = 3\) into the second equation:
\[
2(3) + 5y = 6 \quad \Rightarrow \quad 6 + 5y = 6 \quad \Rightarrow \quad 5y = 0 \quad \Rightarrow \quad y = 0
\]

Solution: \(x = 3, y = 0\)

---

Would you like me to continue solving the other systems (d, e, f), or do you have any questions about these steps?

### Related Questions:
1. How can you solve a system of linear equations using substitution?
2. What is the difference between the substitution and elimination methods for solving systems?
3. How can you check if a system of equations has no solution or infinitely many solutions?
4. How do you solve a system of equations graphically?
5. What are the conditions for a system of linear equations to be consistent?

### Tip:
When solving systems of equations, always aim to simplify the equations first before solving, either by eliminating a variable or using substitution.

Solve the system of equations: a) 3x + y = 18 and 2x - y = 7; b) 4x + 3y = 2 and 5x - 3y = 16; c) 12x - 5y = 36 and 2x + 5y = 6; d) 14x - 7y = 7 and 7y + 3x = 7; e) 4x + 3y = 14 and 5y = 4x - 30; f) 5x - 28 = 6y and 5x + 3y = -19.

Learn how to solve systems of linear equations with two variables. We cover equations such as 3x + y = 18 and 2x - y = 7, along with other systems using elimination and substitution methods. Perfect for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation