Question

$y=ax-5$
$y=x+6$
$y=3x+b$
In the xy-plane, the straight-line graphs of the three equations above each contain the point (p,r), If a and b are constants, what is the value of b ?
(1) a = 2
(2) r = 17

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

The correct option is D Each statement alone is sufficient.

• Since (p,r) is on each of the lines, each of the following three equations is true: (i) r = ap − 5 (ii) r = p + 6 (iii) r = 3p + b Determine the value of b.
  

1. Given that a = 2, Equations (i) and (ii) become r = 2p − 5 and r = p + 6. Subtracting equations gives 0 = p − 11, or p = 11. Now using r = p + 6, it follows that r = 11 + 6 = 17. Finally, using p = 11 and r = 17 in Equation (iii) gives 17 = 3(11) + b, or b = 17 − 33 = −16; SUFFICIENT.
2. Given that r = 17, Equation (ii) becomes 17 = p + 6, and so p = 17 − 6 = 11. Using r = 17 and p = 11, Equation (iii) becomes 17 = 3(11) + b, or b = 17 − 33 = −16; SUFFICIENT.

• The correct answer is D; each statement alone is sufficient.