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

Try this beautiful sum of Co-ordinates based on co-ordinate Geometry from AMC-10A, 2014. You may use sequential hints to solve the problem.

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

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 $-8$

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