Consider the system of equations
y = 3x + 2
y = −x + 14.
a. solve the system of equations.
b. ilene wants to rotate the lines representing this system of equations about their solution and wishes to apply
the matrix
cos(theta) −sin(theta)
sin(theta) cos(theta) to any point a on either of the lines. if ilene is correct, then applying a rotation
to the solution will map the solution to itself. let theta = 90°, and find where ilene’s strategy maps the solution
you found in part (a). what is wrong with ilene’s strategy?
c. jasmine thinks that in order to apply a rotation to some point on either of these two lines, the entire system
needs to be shifted so that the pivot point is translated to the origin. for an arbitrary point a on either of the
two lines, what transformation needs to be applied so that the pivot point is mapped to the origin?
d. after applying your transformation in part (c), apply ilene’s rotation matrix for theta = 90°. show that the pivot
point remains on the origin. what happens to the point (0,2) after both of these transformations?
e. although jasmine and ilene were able to work together to rotate the points around the pivot point, now their
lines are nowhere near the original lines. what transformation brings the system of equations back so that the
pivot point returns to where it started and all other points have been rotated? find the final image of the
point (0,2).
f. summarize your results in parts (a)–(e).