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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

But try the problem first…

Answer: $$-8$$

Source

AMC-10A (2014) Problem 21

Pre College Mathematics

## Try with Hints

First hint

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 ……….

Second Hint

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……..

Final Step

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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