https://r2.erweima.ai/mathbot/image/1725153922778uttxqejz.jpg

The given exercise is a system of two equations with two variables, \(x\) and \(y\). The system is:

\[
\begin{cases} 
x = \frac{4x + 1}{9} = \frac{2y - 5}{3} \\ 
y = \frac{3y + 2}{7} = \frac{x + 18}{10} 
\end{cases}
\]

To solve this system of equations, we need to equate and solve for both variables step by step.

### Step 1: Solving the first equation
Let's start with the first equation:

\[
\frac{4x + 1}{9} = \frac{2y - 5}{3}
\]

To eliminate the denominators, we can cross-multiply:

\[
3(4x + 1) = 9(2y - 5)
\]

Simplify the equation:

\[
12x + 3 = 18y - 45
\]

Rearrange to bring like terms together:

\[
12x - 18y = -48 \implies 2x - 3y = -8 \quad (1)
\]

### Step 2: Solving the second equation
Now, let's solve the second equation:

\[
\frac{3y + 2}{7} = \frac{x + 18}{10}
\]

Cross-multiply to eliminate the denominators:

\[
10(3y + 2) = 7(x + 18)
\]

Simplify the equation:

\[
30y + 20 = 7x + 126
\]

Rearrange to bring like terms together:

\[
7x - 30y = -106 \quad (2)
\]

### Step 3: Solving the system of linear equations
We have two linear equations:

1. \(2x - 3y = -8\)
2. \(7x - 30y = -106\)

To solve this system, we can use either substitution or elimination. I'll use the elimination method.

Multiply the first equation by 7 and the second equation by 2:

\[
\begin{cases} 
14x - 21y = -56 \\
14x - 60y = -212 
\end{cases}
\]

Subtract the first equation from the second:

\[
(-21y + 60y = -56 + 212) \implies 39y = 156 \implies y = 4
\]

### Step 4: Find \(x\) using the value of \(y\)
Substitute \(y = 4\) into the first equation:

\[
2x - 3(4) = -8 \implies 2x - 12 = -8 \implies 2x = 4 \implies x = 2
\]

### Solution
The solution to the system is:

\[
x = 2, \quad y = 4
\]

Would you like more details or have any questions?

### Related Questions:
1. How to solve a system of equations using the substitution method?
2. What are other methods for solving simultaneous equations?
3. Can systems of equations have more than one solution?
4. What happens when the system of equations is dependent or inconsistent?
5. How does cross-multiplication work in solving equations?

**Tip:** When solving systems of equations, always check your solution by substituting the values back into the original equations.

Learn how to solve a system of two equations with two variables step-by-step. This problem involves using the elimination method and cross-multiplication to find the values of x and y. Suitable for high school students studying systems of linear equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation