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If $ax+3y=5$ and $2x+by=3$ represent the same straight line, then what does a+b equal?

I've tried this,

$ax+3y=5$ and $2x+by=3$

Multiply to equal 15 so they equal each other

$$3ax+9y=10x+5by$$

Now what do I do?

How do I find the value of $a$ and $b$? Thanks

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2 Answers 2

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Since the two equations are of the same line -

${a\over2}={3\over b}={5\over3}$

So, $a={10\over3}$ and $b={9\over 5}$

So $a+b={77\over 15}$

Hope this helps!

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    $\begingroup$ How do I get ${a \over 2}$ and ${3 \over b}$? $\endgroup$
    – didgocks
    Commented Feb 7, 2016 at 18:52
  • $\begingroup$ If the equations $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ represent the same line, then - $a_1/a_2$=$b_1/b_2$=$c_1/c_2$ $\endgroup$ Commented Feb 7, 2016 at 18:53
  • $\begingroup$ Oh yes, a silly mistake $\endgroup$
    – didgocks
    Commented Feb 7, 2016 at 18:54
  • $\begingroup$ Haha! Happens to the best of us :D $\endgroup$ Commented Feb 7, 2016 at 18:55
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You are on the right track, but you still need to include the 15 at the end.

$$3ax+9y=10x+5by=15$$

From here you can solve for a and b by matching up the coefficients in front of x and y. In this case, $3ax=10x$ and $9y=5by$ so $a=\frac{10}{3}$ and $b=\frac{9}{5}$

So their sum is $$a+b=\frac{50}{15}+\frac{27}{15}=\frac{77}{15}$$

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