Simultaneous Linear Equations

To determine two unknowns, two independent linear equations are usually required. A pair of simultaneous linear equations may be written as

where \(a,b,c,d,e\) and \(f\) are constants. The solution is the common pair of values of \(x\) and \(y\) that makes both equations true at the same time.

For example, \(x=2\) and \(y=3\) solve \(x+y=5\) and \(2x-y=1\), because the pair satisfies both equations.

• Graphical method: draw the two lines and read their point of intersection.

• Elimination method: combine the equations to remove one unknown.

• Substitution method: express one unknown in terms of the other, then substitute.

After solving, substitute the values into both original equations to check the answer.