Find the values of x and y from each of the following
(i) `(x+y,x-2y)=(7,1)`
(ii) `(2x,x+3y)=(4,5)`

Answer

To solve the given equations step by step, we will treat each ordered pair separately.

### Part (i): Solve `(x+y, x-2y) = (7, 1)`

1. **Set up the equations**:
From the ordered pair equality, we can write:
- \( x + y = 7 \) (1)
- \( x - 2y = 1 \) (2)

2. **Substitute equation (1) into equation (2)**:
We can express \( x \) in terms of \( y \) using equation (1):
\[ x = 7 - y \]
Now, substitute this expression for \( x \) into equation (2):
\[ (7 - y) - 2y = 1 \]

3. **Simplify the equation**:
Combine like terms:
\[ 7 - y - 2y = 1 \implies 7 - 3y = 1 \]

4. **Solve for \( y \)**:
Rearranging gives:
\[ -3y = 1 - 7 \implies -3y = -6 \implies y = 2 \]

5. **Substitute \( y \) back to find \( x \)**:
Now substitute \( y = 2 \) back into equation (1):
\[ x + 2 = 7 \implies x = 5 \]

### Solution for Part (i):
Thus, the values are \( x = 5 \) and \( y = 2 \).

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### Part (ii): Solve `(2x, x + 3y) = (4, 5)`

1. **Set up the equations**:
From the ordered pair equality, we can write:
- \( 2x = 4 \) (3)
- \( x + 3y = 5 \) (4)

2. **Solve for \( x \)**:
From equation (3):
\[ 2x = 4 \implies x = \frac{4}{2} = 2 \]

3. **Substitute \( x \) back into equation (4)**:
Now substitute \( x = 2 \) into equation (4):
\[ 2 + 3y = 5 \]

4. **Solve for \( y \)**:
Rearranging gives:
\[ 3y = 5 - 2 \implies 3y = 3 \implies y = 1 \]

### Solution for Part (ii):
Thus, the values are \( x = 2 \) and \( y = 1 \).

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### Summary of Solutions:
- For part (i): \( x = 5 \), \( y = 2 \)
- For part (ii): \( x = 2 \), \( y = 1 \)

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