Simultaneous Equations and Graphs

Simultaneous equations are a set of equations with multiple variables that are all satisfied by the same values. The aim is to find these common values.
You can solve a system of simultaneous equations either by substitution, by elimination, or using graphical methods.
Substitution Method: Solve one of the equations for one variable and then substitute this expression into the other equation. Simplify and solve for the remaining variable.
Elimination Method: To use this method, either add or subtract the equations in a way that eliminates one of the variables, making it possible to easily solve for the other.
Graphical Method: If both equations are in the y = format, the solution will be where the graphs of the two equations intersect on a graph. The x-coordinate and the y-coordinate at this point are the solutions for the two equations.
When plotting linear equations on graphs, remember that the equation y = mx + c represents a straight line. The number ‘m’ is the gradient (slope) of the line, and the number ‘c’ is where the line crosses the y-axis (y-intercept).
Simultaneous equations can have one solution, no solution, or an infinite number of solutions. If both equations represent the same line, the system will have an infinite number of solutions. If the lines are parallel, the system will have no solutions.
Two simultaneous linear equations always represent straight lines. If both lines intersect, this point of intersection is the solution to the system of equations.
Quadratic simultaneous equations involve at least one equation that is quadratic. You may need to use substitution to solve these types of equations.
The simultaneous equations involving circles might require knowledge about the equation of a circle, which is (x-a)^2 + (y-b)^2 = r^2 where (a, b) is the center of the circle and ‘r’ is the radius.
Developing strong problem-solving skills is especially important in this topic. Some problems might require setting up a system of equations from a given word problem.
Practice is key. Solve as many problems as you can, so you become familiar with different types of simultaneous equations and the various techniques to solve them.
After obtaining the solutions, always check them by substituting them back into the original equations to confirm they hold true.