Question

Lines $2x-by+5=0$ and $ax+3y=2$are parallel. Find the relation connecting $a$ and $b$.

Answer 1

Step1: Calculation of slope of line $2x-by+5=0$.

The slope-intercept form of the equation of a line is given by $y=mx+c$, where $m$ is the slope and $c$ is the $Y$-intercept.

Simplify the equation $2x-by+5=0$ to convert it into slope-intercept form.

$$\begin{gathered} 2x-by+5=0 \\ \Rightarrow -by=-(2x+5)(Subtracting2x+5tobothsides) \\ \therefore y=\frac{2}{b}x+\frac{5}{b}(dividingbothsidesby-b)-equation\left(1\right) \end{gathered}$$

Comparing equation (1) with equation $y=mx+c$, we get the slope of the line $2x+by+5=0$ as $m_1=\frac{2}{b}$.

Step2: Calculation of slope of line $ax+3y=2$.

Simplify the equation $ax+3y=2$ to convert it into slope-intercept form.

$$\begin{gathered} ax+3y=2 \\ \Rightarrow 3y=-ax+2(subtractingaxfrombothsides) \\ \therefore y=-\frac{a}{3}x+\frac{2}{3}(dividingbothsidesby3)-equation\left(2\right) \end{gathered}$$

Comparing equation (2) with equation $y=mx+c$, we get the slope of the line $ax+3y=2$ as $m_2=-\frac{a}{3}$.

Step3: Calculate the relation connecting $a$ and $b$.

Since, the lines $2x-by+5=0$ and $ax+3y=2$ are parallel, their slopes will be equal.

$$\begin{gathered} \Rightarrow \frac{2}{b}=-\frac{a}{3} \\ \Rightarrow 6=-ab(multiplyingbothsidesby3b) \\ \therefore ab=-6(multiplyingbothsidesby-1andrearranging) \end{gathered}$$

Hence, the relation connecting $a$ and $b$ is $ab=-6$.