1. Substitution Method:
   • Step 1: Solve one of the equations for one variable in terms of the other. For example, if you have the equations (y = 2x + 3) and (x + y = 10), you can express (y) in terms of (x).
   • Step 2: Substitute this expression into the other equation. This will give you an equation with one variable.
   • Step 3: Solve for the remaining variable.
   • Step 4: Substitute back to find the value of the other variable.
2. Elimination Method:
   • Step 1: Align the equations vertically. For example, if you have (3x + 2y = 12) and (2x - y = 3), write them one above the other.
   • Step 2: Multiply one or both equations by a constant to make the coefficients of one variable the same or opposites.
   • Step 3: Add or subtract the equations to eliminate one variable.
   • Step 4: Solve for the remaining variable and substitute back to find the other variable.
3. Graphical Method:
   • Step 1: Graph both equations on the same coordinate plane.
   • Step 2: Identify the point where the two lines intersect. This point represents the solution to the system of equations.
   • Note: This method is useful for visualizing the solution but may not be precise for complex equations.

Consider the system of equations:

1. (x + y = 5)
2. (2x - y = 1)Using the Substitution Method:

• From the first equation, express (y): (y = 5 - x).
• Substitute into the second equation: (2x - (5 - x) = 1) → (3x - 5 = 1) → (3x = 6) → (x = 2).
• Substitute (x) back into the first equation: (2 + y = 5) → (y = 3).
• Solution: (x = 2), (y = 3).Using the Elimination Method:
• Align the equations:
  1. (x + y = 5) (multiply by 1)
  2. (2x - y = 1) (multiply by 1)
• Add the equations: ((x + y) + (2x - y) = 5 + 1) → (3x = 6) → (x = 2).
• Substitute back to find (y): (2 + y = 5) → (y = 3).
• Solution: (x = 2), (y = 3).By using these methods, you can effectively solve systems of simultaneous equations in various contexts. Each method has its advantages, and the choice of method may depend on the specific problem or personal preference.

1. Substitution Method:
   • Step 1: Solve one of the equations for one variable in terms of the other. For example, if you have the equations (y = 2x + 3) and (3x + y = 9), you can substitute (y) in the second equation.
   • Step 2: Substitute this expression into the other equation and solve for the remaining variable.
   • Step 3: Once you find the value of one variable, substitute it back into one of the original equations to find the other variable.
2. Elimination Method:
   • Step 1: Align the equations so that like terms are in columns. For example, if you have (2x + 3y = 6) and (4x - 3y = 12), write them one above the other.
   • Step 2: Add or subtract the equations to eliminate one variable. You may need to multiply one or both equations by a number to align coefficients.
   • Step 3: Solve for the remaining variable, then substitute back to find the other variable.
3. Graphing Method:
   • Step 1: Graph each equation on the same coordinate plane. The point where the lines intersect represents the solution to the system.
   • Step 2: If the lines intersect at a single point, that point is the solution. If the lines are parallel, there is no solution. If they overlap, there are infinitely many solutions.

Consider the system of equations:

• (x + y = 5) (1)
• (2x - y = 1) (2)Using Substitution:

1. From (1), express (y): (y = 5 - x).
2. Substitute into (2): (2x - (5 - x) = 1) → (3x - 5 = 1) → (3x = 6) → (x = 2).
3. Substitute (x = 2) back into (1): (2 + y = 5) → (y = 3).
4. Solution: (x = 2), (y = 3).By using these methods, you can effectively solve systems of equations to find the values of the unknown variables. Each method has its advantages, and the choice of method may depend on the specific equations you are working with.