Simultaneous Equations

What are Simultaneous Equations?

• Simultaneous equations are a set of two or more equations that share the same variables.

• The goal is to find values for the variables that satisfy all the equations simultaneously (at the same time).

Example:

• Equation 1: 2x + y = 7

• Equation 2: x - y = 1

Methods for Solving Simultaneous Equations

1. Substitution Method:

   • Step 1: Solve one equation for one variable in terms of the other.

   • Step 2: Substitute the expression from step 1 into the other equation.

   • Step 3: Solve the resulting equation for the remaining variable.

   • Step 4: Substitute the value found in step 3 back into either of the original equations to find the value of the other variable.   

2. Elimination Method:

   • Step 1: Multiply one or both equations by constants to make the coefficients of one of the variables the same or opposite.

   • Step 2: Add or subtract the equations to eliminate one of the variables.

   • Step 3: Solve the resulting equation for the remaining variable.

   • Step 4: Substitute the value found in step 3 back into either of the original equations to find the value of the other variable.   

3. Graphical Method:

   • Step 1: Graph each equation on the same coordinate plane.

   • Step 2: The point of intersection of the two lines represents the solution to the system of equations.

Applications of Simultaneous Equations

• Real-world problems: Solving problems involving mixtures, distances, speeds, ages, and more.

• Systems of linear equations: Used in various fields like economics, engineering, and physics.