Algebra Know-It-ALL

We’ve seen how pairs of linear equations can be solved as two-by-two systems. What if the

equations are more complicated? In this chapter, we’ll solve some two-by-two systems in

which one or both of the equations are quadratic or cubic.

Linear and Quadratic

When we want to solve a pair of two-variable equations together as a system, we can go

through these steps in order.

• Decide which variable to call independent, and which variable to call dependent.


• Morph both equations so they express the dependent variable in terms of the indepen-
  dent variable.


• Mix the independent-variable parts of the equations to get an equation in a single
  variable.


• Find the root or roots of that equation.


• Plug the root or roots into one of the morphed original equations, and calculate the
  corresponding value or values of the dependent variable.


• Express the solutions as ordered pairs with the independent variable listed first.

First, we morph

Let’s try an example. Consider these two equations, the first of which is linear and the second

of which is quadratic:

2 x−y+ 1 = 0

and

y=x^2 +x− 5

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CHAPTER

27 More Two-by-Two Systems

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