Solving Simultaneous Linear Equations in 2 Unknowns Revision Content

Understanding Simultaneous Linear Equations

Simultaneous linear equations are algebraic expressions that contain two or more unknowns which are required to satisfy all the given equations.
When dealing with 2 unknowns, our goal is to find the pair of values that satisfy both equations.
They are called ‘simultaneous’ because the solution must satisfy all equations at the same time.

Simultaneous linear equations can be represented graphically as straight lines in a Cartesian plane.
Where these lines intersect, or coincide, is the solution (i.e., the values of x and y) that satisfies both equations.

The substitution method involves rearranging one of the equations to find one variable in terms of the other. This new equation is then substituted into the second equation to find the value of the second variable.
Once one variable’s value is found, it can be substituted back into one of the original equations to find the other variable’s value.

The elimination method is based on the idea of adding or subtracting the equations from each other in order to eliminate one of the variables.
By performing the same operation to both sides of an equation, we preserve its equality. With a little foresight, this can be used to remove one of the variables, resulting in an equation with only one unknown.
Once one variable’s value is determined, this can be substituted back into one of the original equations to solve for the other variable.

The linear combination method is another version of the elimination method. The difference is that, before adding or subtracting the equations, we may multiply each equation by a suitable number so that the coefficients of one of the variables are the same (or additive inverses), allowing it to be eliminated.

It is important to double-check your solution by substituting the variable values back into both original equations to ensure they are indeed correct.

Understanding how to solve simultaneous equations is fundamental for the study of algebra and is important in many applications including physics, business, and economics, where multiple conditions must be satisfied simultaneously.

Becoming familiar with the mathematical language used to express these equations can often make the task of identifying the most effective solution method easier.
Recognising when an equation is in a standard form (Ax + By = C), where the coefficients and constants are integers, can often simplify the process of identifying how best to add or subtract the equations.

If the lines representing the equations on the Cartesian plane are parallel, there is no solution as the lines do not intersect.
If both equations represent the same line, there are infinitely many solutions as every point on the line solves both equations.