1.5 Equivalent Linear Systems

You can solve the following system of linear equations by graphing:x + 2y = 104x – y = -14Point of intersection seems to be (-2, 6) What other systems of linear equations have the same solution?1. We can add the equations.2. We can subtract the equations.

Add/Subtract EquationsAdd x + 2y = 10+ 4x – y = -14 5x + y = -4

Subtract x + 2y = 10- 4x – y = -14 -3x + 3y = 24

5x + y = -4 and -3x + 3y = 24 have the same point of intersection as the original equations which created them.

These are all equivalent systems of linear equations.

Equivalent System of Linear Equations

Example #2Multiply x + 2y = 10 by 4 and 4x – y = -14 by -2. Graph the original and two new equations on the same axes.4x + 8y = 40 and -8x + 2y = 28

The lines are exactly the same as the original ones (because both sides got multiplied by the same number).Both have same point of intersection.

Note…Just as we can add 2 equations and have the point of intersection be unchanged, and just as we can multiply both sides of an equation by the same number and have the point of intersection be unchanged, we can perform both these operations at the same time and still have the point of intersection be unchanged.

In Summary…• When you add/subtract the equations in a linear

system:• New linear system has same solutions as original

system (same point of intersection)• Graphs of new equations are different than

graphs of original equations:• When you multiply an equation with a constant other

than 0:• This does not change the graph of the equation• New linear system has equations with same point

of intersection as original equations