Try this beautiful problem from Algebra based on Sum of Co-ordinates

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

Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$-axis at the same point. What is the sum of all possible $x$-coordinates of these points of intersection?

- \(16\)
- \(12\)
- \(-8\)
- \(-4\)

**Key Concepts**

Geometry

Co-ordinate

## Check the Answer

Answer: \(-8\)

AMC-10A (2014) Problem 21

Pre College Mathematics

## Try with Hints

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 \(-8\)

## Other useful links

- https://www.cheenta.com/area-of-region-in-a-circle-amc-10a-2011-problem-18/
- https://www.youtube.com/watch?v=d9WXHQFqbWs&t=4s