Students are given 3 minutes for each multiple-choice question and 5 minutes for each free-response question on a test. there are 15 questions on the test, and students are given 51 minutes to take it. the system of equations shown can be used to find the number of multiple-choice questions, m, and the number of free-response questions, f, on the test. m + f = 15 3m + 5f = 51 how many multiple-choice questions are on the test?

M + f = 15multiply by -5
3m + 5f = 51

-5m - 5f = - 75 (result of multiplying by -5)
3m + 5f = 51
add
-2m = - 24
m = -24/-2
m = 12 <=== 12 multiple choice questions

Intersecting secant-tangent theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.so to break it down: 1. pq acts as the tangent and ps as the secant with an intersection at p.sr = 21rp = 3x + 3pq = 4x + 4following to equation: tangent squared = the two segments multiplied by each otherso (4x + 4) ^ 2 = (3x + 3) • (21) simplifies to 16x^2 - 31x - 47 = 0 or (16x - 47)(x + 1) x = -1 or x = 47/162. pq acts as the tangent and ps as the secant with an intersection at p (again! )sr = 16rp = xpq = 15(15) ^ 2 = 16 • x225 = 16xx = 225/163. the interesting chords are proportional to each other so a ratio is possible to set up: ap dp 10 3 + x = = —— = pc pb 8 xcross multiple to get 10x = (3 + x)(8)x = 12