**explanation of the algebraic methods for solving a pair of linear equations:**

**1. Substitution Method:**

**Steps:**

**Solve one equation for one variable:**Choose an equation and rearrange it to isolate one variable in terms of the other.**Substitute:**Substitute the expression you found for that variable into the other equation. This will create an equation with only one variable.**Solve:**Solve the resulting equation for the remaining variable.**Back-substitute:**Plug the value you found for that variable back into the first equation to solve for the other variable.

**Example:**

Given the equations:

2x + 3y = 7 x – y = 1

Solve for x using substitution:

- Rearrange the second equation to isolate x: x = y + 1
- Substitute this expression for x in the first equation: 2(y + 1) + 3y = 7
- Simplify and solve for y: 2y + 2 + 3y = 7 => 5y = 5 => y = 1
- Substitute y = 1 back into x = y + 1 to find x: x = 1 + 1 => x = 2

**2. Elimination Method:**

**Steps:**

**Make coefficients equal:**Multiply one or both equations by appropriate numbers so that one variable has the same coefficient (but opposite signs) in both equations.**Add or subtract equations:**Add or subtract the equations to eliminate one variable.**Solve for remaining variable:**Solve the resulting equation for the remaining variable.**Back-substitute:**Plug the value you found for that variable back into one of the original equations to solve for the other variable.

**Example:**

Given the same equations as before:

2x + 3y = 7 x – y = 1

Solve for x using elimination:

- Multiply the second equation by 2: 2x – 2y = 2
- Add the first and second equations: (2x + 3y) + (2x – 2y) = 7 + 2 => 4x + y = 9
- Solve for x: 4x = 9 – y => x = (9 – y)/4
- Substitute y = 1 (from the previous example) to find x: x = (9 – 1)/4 => x = 2

**Choosing the Method:**

**Substitution:**Often simpler when one equation is already solved for one variable.**Elimination:**Convenient when coefficients of one variable are already additive inverses or easily made so.

**Consider factors like personal preference and the ease of manipulating the given equations when selecting a method.**