OG The straight-line graphs of the three equations

AbeNeedsAnswers wrote: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

D

We can begin by substituting p and r for x and y, respectively, in the three given equations.

1) r = ap – 5

2) r = p + 6

3) r = 3p + b

Statement One Alone:

a = 2

We can substitute 2 for a in the equation r = ap – 5. Thus, we have:

r = 2p – 5

Next we can set equations 1 and 2 equal to each other.

2p – 5 = p + 6

p = 11

Since p = 11, we see that r = 11 + 6 = 17

Finally, we can substitute 11 for p and 17 for r in equation 3. This gives us:

17 = 3(11) + b

17 = 33 + b

-16 = b

Statement one alone is sufficient to answer the question.

Statement Two Alone:

r = 17

We can substitute r into all three equations and we have:

1) 17 = ap – 5

2) 17 = p + 6

3) 17 = 3p + b

We see that p = 11. Now we can substitute 11 for p in equation 3 to determine a value for b.

17 = 3(11) + b

-16 = b

Statement two alone is also sufficient to answer the question.

Answer: D

Hi AbeNeedsAnswers,

We're given the equations for 3 lines (and those equations are based on 4 unknowns: 2 variables and the 2 'constants' A and B):
Y = (A)(X) - 5
Y = X + 6
Y = 3X + B
We're told that the three lines all cross at one point on a graph (p,r). We're asked for the value of B. While this question looks complex, it's actually built around a 'system' math "shortcut" - meaning that since we have 3 unique equations and 4 unknowns, we just need one more unique equation (with one or more of those unknowns) and we can solve for ALL of the unknowns:

1) A =2

With this information, we now have a 4th equation, so we CAN solve for B.
Fact 1 is SUFFICIENT

2) R = 17

This information tell us the x co-ordinate where all three lines will meet, so it's the equivalent of having X=17 to work with. This 4th equation also allows us to solve for B.
Fact 2 is SUFFICIENT

Final Answer: D

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