Week 12 – Precalc 11 – John Carlo's Blog

One of the things we learned this week in Precalc 11 is how to solve Systems of Equations Algebraically.

TYPES OF SYSTEMS

Linear-Linear System is a system consisting of two lines that can have 0, 1 and infinite solutions.
Linear-Quadratic System is a system consisting of a line and a parabola that can have 0, 1 and 2 solutions.
Quadratic-Quadratic System is a system consisting of a two parabolas that can have 0, 1, 2 and infinite solutions.

Solving Systems of Equations Algebraically

Example 1

1.) $y = -2x^2 + 9$
2.) $y = 2x^2 - 8x + 9$
The first step in solving systems using the substitution method is to make one equation equal to x or y, in this case both equations are equal to y. Next, we could either replace y in the first equation using the second second or replace y in the second equation using the first. I will replace y in the second equation.
$-2x^2+9=2x^2-8x+9$
The next step would be to make the whole equation equal to 0.
$0=4x^2-8x$
Next, factor.
$0=4x(x-2)$
Then we determine the zeros.
$x_1=0$
$x_2=2$
So we know the x coordinates of the solutions (0, __ ) and (2, __ ) to find out what the y values are, we simply replace the x’s in any of the equations. I will replace the x’s in the first equation.
$y = -2(0)^2 + 9$
$y = 9$
First solution is (0, 9)
$y = -2(2)^2 + 9$
$y = -8 + 9$
$y = 1$
Second solution is (2, 1)