Solve the following system of equations algebraically:
y = x2 - 6x + 9
y = -9x + 19

{(-5,64), (2,1)}

linear equation: y = -9x + 19
quadratic equation: y = x² - 6x + 9

Substitute the y in the quadratic equation by the its value in the linear equation.

-9x + 19 = x² - 6x + 9
      -  19               - 19    *subtract 19 to both sides
-9x         = x² - 6x -10
+9x              + 9x            *add 9x to both sides
0            = x² + 3x - 10
0            = (x + 5) (x - 2) *Factor

Set each factor = 0 and solve
x + 5 = 0    ;      x - 2 = 0
x = -5         ;      x = 2

Find the corresponding value of y using the linear equation.
y = -9x + 19

x = -5                              x = 2
y = -9(-5) + 19               y = -9(2) + 19
y = 45 + 19                    y = -18 + 19
y = 64                            y = 1

(-5,64)                              (2,1)

Check each value on each equation.
y = x² - 6x + 9
(-5,64)                                                  (2,1)
64 = (-5)² - 6(-5) + 9                          1 = 2² - 6(2) + 9
64 = 25 + 30 + 9                                1 = 4 - 12 + 9
64 = 64                                               1 = 1

y = -9x + 19
64 = -9(-5) + 19                                  1 = -9(2) + 19
64 = 45 + 19                                      1 = -18 + 19
64 = 64                                              1 = 1

{(-5,64), (2,1)}

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