Question

Express the following linear equations in the form \[ax+by+c=0\] and indicate the values of a, b, and c in each case:
i) $2x+3y=9.3\overline{5}$
ii) $x-\dfrac{y}{5}-10=0$
iii) $-2x+3y=6$
iv) $x=3y$
v) $2x=-5y$
vi) $3x+2=0$
vii) $y-2=0$
viii) $5=2x$

Answer 1

Hint: First, move all parts to the left side of the equation. Then, compare it with the equation $ax+by+c=0$. After that equate it with the coefficient of $x$, $y$ and constant part. The value of a, b and c is the desired result. We will follow the same approach for every part.

Complete step by step answer:
(i) Given: $2x+3y=9.3\overline{5}$
Move $9.3\overline{5}$ on the right side,
$\Rightarrow 2x+3y-9.3\overline{5}=0$
Compare the equation with $ax+by+c=0$,
$\Rightarrow a=2$
$\Rightarrow b=3$
$\Rightarrow c=9.3\overline{5}$

Hence, the values are $a=2$, $b=3$ and $c=9.3\overline{5}$.

(ii) Given: $x-\dfrac{y}{5}-10=0$
Compare the equation with $ax+by+c=0$,
$\Rightarrow a=1$
$\Rightarrow b=-\dfrac{1}{5}$
$\Rightarrow c=-10$

Hence, the values are $a=1$, $b=-\dfrac{1}{5}$ and $c=-10$.

(iii) Given: $-2x+3y=6$
Move 6 on the right side,
$\Rightarrow -2x+3y-6=0$
Compare the equation with $ax+by+c=0$,
$\Rightarrow a=-2$
$\Rightarrow b=3$
$\Rightarrow c=-6$

Hence, the values are $a=-2$, $b=3$ and $c=-6$.

(iv) Given: $x=3y$
Move 3y on the right side,
$\Rightarrow x-3y=0$
Compare the equation with $ax+by+c=0$,
$\Rightarrow a=1$
$\Rightarrow b=-3$
Since there is no constant part,
$\Rightarrow c=0$

Hence, the values are $a=1$, $b=-3$ and $c=0$.

(v) Given: $2x=-5y$
Move 5y on the right side,
$\Rightarrow 2x+5y=0$
Compare the equation with $ax+by+c=0$,
$\Rightarrow a=2$
$\Rightarrow b=5$
Since there is no constant part,
$\Rightarrow c=0$

Hence, the values are $a=2$, $b=5$ and $c=0$.

(vi) Given: $3x+2=0$
Compare the equation with $ax+by+c=0$,
$\Rightarrow a=3$
$\Rightarrow c=2$
Since there is no value of $y$,
$\Rightarrow b=0$

Hence, the values are $a=3$, $b=0$ and $c=2$.

(vii) Given: $y-2=0$
Compare the equation with $ax+by+c=0$,
$\Rightarrow b=1$
$\Rightarrow c=-2$
Since there is no value of x,
$\Rightarrow a=0$

Hence, the values are $a=0$, $b=1$ and $c=-2$.

(viii) Given: $5=2x$
Move 2x on the right side,
$\Rightarrow -2x+5=0$
Compare the equation with $ax+by+c=0$,
$\Rightarrow a=-2$
$\Rightarrow c=5$
Since there is no value of y,
$\Rightarrow b=0$

Hence, the values are $a=-2$, $b=0$ and $c=5$.

Note:
An equation is said to be a linear equation in two variables if it is written in the form of \[ax+by+c=0\], where a, b & c are real numbers and the coefficients of x and y, i.e. a and b respectively, are not equal to zero.
The solution for such an equation is a pair of values, one for x and one for y which further makes the two sides of an equation equal.