Question

Which pair of points lie on the same side of $3x-8y-7=0$

• A. $\left(0,-1\right)and\left(0,0\right)$
• B. $\left(4,-3\right)and\left(0,1\right)$
• C. $\left(-3,-4\right)and\left(1,2\right)$
• D. $\left(-1,-1\right)and\left(3,7\right)$

Answer 1

The correct option is D

$\left(-1,-1\right)and\left(3,7\right)$

Explanation for the correct answer:

Option D:

The points will be on the same side of the line if both the values after substitution are either greater or lesser than zero.

The given line is $3x-8y-7=0$.

Let $f\left(x,y\right)=3x-8y-7$

Substituting $\left(-1,-1\right)and\left(3,7\right)$ in $f\left(x,y\right)=3x-8y-7$.

$f\left(-1,-1\right)=3\left(-1\right)-8\left(-1\right)-7$

$\Rightarrow$$f\left(-1,-1\right)=-3+8-7$

$\Rightarrow$$f\left(-1,-1\right)=-2<0$

$f\left(3,7\right)=3\left(3\right)-8\left(7\right)-7$

$\Rightarrow$ $f\left(3,7\right)=9-56-7$

$\Rightarrow$ $f\left(3,7\right)=-54<0$

Since both the values are less than zero, the points $\left(-1,-1\right)and\left(3,7\right)$ are on the same side.

Hence, the correct option is D

Explanation for the wrong option:

Option A: $\left(0,-1\right)and\left(0,0\right)$

$f\left(0,-1\right)=3\left(0\right)-8\left(-1\right)-7$

$\Rightarrow$ $f\left(0,-1\right)=0+8-7$

$\Rightarrow$ $f\left(0,-1\right)=1>0$

$f\left(0,0\right)=3\left(0\right)-8\left(0\right)-7$

$\Rightarrow$ $f\left(0,0\right)=-7<0$

One value is greater than zero and the other value is lesser than zero. So the points lie on the opposite sides.

Option B: $\left(4,-3\right)and\left(0,1\right)$

$f\left(4,-3\right)=3\left(4\right)-8\left(-3\right)-7$

$\Rightarrow$$f\left(4,-3\right)=12+24-7$

$\Rightarrow$$f\left(4,-3\right)=29>0$

$f\left(0,1\right)=3\left(0\right)-8\left(1\right)-7$

$\Rightarrow$ $f\left(0,1\right)=0-8-7$

$\Rightarrow$ $f\left(0,1\right)=-15<0$

One value is greater than zero and the other value is lesser than zero. So the points lie on the opposite sides.

Option C: $\left(-3,-4\right)and\left(1,2\right)$

$f\left(-3,-4\right)=3\left(-3\right)-8\left(-4\right)-7$

$\Rightarrow$$f\left(-3,-4\right)=-9+32-7$

$\Rightarrow$$f\left(-3,-4\right)=16>0$

$f\left(1,2\right)=3\left(1\right)-8\left(2\right)-7$

$\Rightarrow$ $f\left(1,2\right)=3-16-7$

$\Rightarrow$ $f\left(1,2\right)=-20$

One value is greater than zero and the other value is lesser than zero. So the points lie on the opposite sides.

Hence, the correct option is D