Systems of Equations

ACT Math · Axiom Academy

Two equations, one shared answer — solve it algebraically with substitution or elimination, and see why it's really just the point where two lines cross. 1. A Solution Is Where the Lines Cross Every linear equation in two variables graphs as a straight line. When you have two equations, you have two lines on the same plane — and the system's solution is the one point that lies on both lines at once: their intersection . The system we'll solve throughout this lesson Substitution works best when one equation already has a variable isolated (or can be isolated in one step). Solve one equation for one variable, then plug that expression into the other equation — you're left with one equation in one variable. Solve x+y=5 and 2x-y=4 . From the first equation, y=5-x . Substitute into the second: 2x-(5-x)=4 , so 3x-5=4 , giving x=3 . Then y=5-3=2 . Solution: (3,2) . Elimination works best when the coefficients on one variable are already equal or opposite (or can be made that way by multiplying an equation). Add or subtract the two equations so that one variable's terms cancel — you're left with one equation in one variable, just like substitution. Solve the same system: x+y=5 and 2x-y=4 . The y -coefficients are already opposite ( +1 and -1 ), so add the equations directly: (x+y)+(2x-y)=5+4 , giving 3x=9 , so x=3 . Substitute back: 3+y=5 , so y=2 . Same solution: (3,2) . 4. Special Cases and Choosing Your Method

This is the written version of the interactive lesson above. See the full ACT Math course.