Solve this system of equations algebraically:
y + x = 19 – x2
x + y = 80
1. isolate one variable in the system of equations. y = –x2 – x + 19
y = –x + 80
2. use substitution to create a one-variable equation. –x + 80 = –x2 – x + 19
3. solve to determine the unknown variable in the equation.

the system has
real number solution(s).

the mistake in the pattern is that :

b. the decimal point was moved one too many place values each time

the correct pattern could be made by :

b. moving the decimal point to the right the same numbers of place values as the exponents

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.