Try this beautiful problem from Algebra based on Sum of Co-ordinates
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?
Geometry
Co-ordinate
Answer: \(-8\)
AMC-10A (2014) Problem 21
Pre College Mathematics
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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