Sum of Co-ordinates | AMC-10A, 2014 | Problem 21

The given equations are $y=ax+5$\(\Rightarrow x=\frac{-5}{a}\)…..(1)

$y=3x+b$\(\Rightarrow x=\frac{-b}{3}\)…………(2)

Since two lines intersect the $x$-axis at the same point,then at first we have to find out the common point on x -axis………

Now the intercept form of the given two equations will be

\(\frac{x}{(-5/a)} +\frac{y}{5}=1\) ,Therefore the straight line intersect x-axis at the point (\(\frac{-5}{a},0\))

\(\frac{x}{(-b/3)}+\frac{y}{b}=1\) Therefore the straight line intersect x-axis at the point (\(\frac{-b}{3},0)\).

Can you now finish the problem ……….

Since two lines intersect the $x$-axis at the same point so we may say that

(\(\frac{-5}{a},0\))=(\(\frac{-b}{3},0)\)

\(\Rightarrow ab=15\)

The only possible pair (a,b) will be \((1,15),(3,5),(5,3),(15,1)\)

can you finish the problem……..

Now if we put the values \((1,15),(3,5),(5,3),(15,1)\) in (1) & (2) we will get \(-5\),\(\frac{-5}{3}\),\(-1\),\(\frac{1}{3}\)

Therefore the sun will be \(-8\)