Algebra Know-It-ALL

Next, we check (−2,−5) in the second original equation:

− 6 x+ 2 y= 2

− 6 × (−2)+ 2 × (−5)= 2

12 + (−10)= 2

2 = 2

Finishing up, we check (−3,−8) in the second original equation:

− 6 x+ 2 y= 2

− 6 × (−3)+ 2 × (−8)= 2

18 + (−16)= 2

2 = 2

Are you confused (or bemused)?

Think back to the notion of multiplicity for the roots of certain quadratic, cubic, and higher-degree equa-
tions. Do you wonder if the same concept applies to the solutions of two-by-two systems when at least one
of the equations is of degree 2 or more? Well, it does! If the equation we create by mixing has a root with
multiplicity of 2 or more, the corresponding solution of the whole system has the same multiplicity. You’ll
see this happen as you work out the last two practice exercises at the end of this chapter.

Here’s a challenge!

Solve these cubic equations as a two-by-two system:

y= 5 x^3 + 3 x^2 + 5 x+ 7

and

y= 2 x^3 +x^2 + 2 x+ 5

Solution

Solving this problem requires some keen intuition, a lot of trial and error, or both. We are lucky in one
respect, at least: These equations are already functions of x, so we have no morphing to do. We can directly
mix the right sides to get

5 x^3 + 3 x^2 + 5 x+ 7 = 2 x^3 +x^2 + 2 x+ 5

Let’s subtract the entire right side of this equation, as a single quantity, from both sides. That changes each
term on the left and sets the right side equal to 0, giving us

3 x^3 + 2 x^2 + 3 x+ 2 = 0

Enter the Cubic 459