Simultaneous Equations

Simultaneous equations are a set of equations with multiple variables, where the solutions must satisfy all equations.
Utilize algebraic methods, including substitution and elimination, to solve simultaneous equations.
For equations with two variables, visual interpretation lies in the intersection of two lines on a graph.

Substitution Method

Substitution is often used when one of the equations makes it convenient to express one variable in terms of the other.
Substitute the expression from one equation into the other and solve for one variable.
Substitute this solution back into one of the original equations to find the other variable.

The Elimination method involves manipulating the equations so that adding or subtracting them eliminates one variable, allowing for solving the remaining variable directly.
Multiply each equation by a suitable number so that the coefficients on x or y are the same in both equations.
Add or subtract the equations to eliminate one variable. Solve the remaining equation for the other variable.
Substitute the solution into one of the original equations and solve for the other variable.

Simultaneous equations can also be represented and solved using matrices.
Create a matrix where each row corresponds to a given equation, and each column represents a variable.
Utilize matrix manipulation techniques like row swapping, row addition or scalar multiplication to solve the system of equations.

Check your answers by substitifying them back into the original equations. If both left-hand and right-hand sides equal, your solutions are correct.
Carefully manage the algebraic signs when working with equations.
Remember even though algebraic methods (Substitution and Elimination) are useful, not every set of simultaneous equations can be solved this way. Some equations require graphical methods or matrices.
Understand the drawbacks and limitations of every method for certain types of problems.

Remember, the key to mastering simultaneous equations is practice. Work on a variety of problems to improve your understanding and skill.