Is this a linear equation #5x+6y=3x-2#?

The generalized linear equation in two variables #x# & #y# is given as

#ax+by+c=0#

The above equation shows a straight line on #XY# plane.

#5x+6y=3x-2#

or #2x+6y+2=0#

or #x+3y+1=0#

The above equation is in form of linear equation: #ax+by+c=0#

#" "#
Linear Equations are referred to as the equations of degree one,

where the variable's exponent ( or Power) is the value one.

Slope-Intercept Form:

This is the most common form of a linear equation.

It takes the form:

#color(red)(y=mx+b#, where

#color(red)(m# is the Slope or the Gradient

#color(red)(b# is the y-intercept

General Form:

IF there are two variables in a linear equation it takes the form:

#color(red)(Ax+By=C#

To find the x-intercept, set #color(blue)(y=0#

To find the y-intercept, set #color(blue)(x=0#

#color(blue)("Note :"#

For both of the above forms, we can construct a data table and create a graph.

Now, we will get back to the problem given to us:

#color(red)(5x+6y = 3x-2#

This is a linear equation with two variables

We can combine like terms and simplify the equation as
follows:

#rArr 5x+6y=3x-2#

Subtract #color(red)(3x# from both sides of the equation to balance the equation.

#rArr 5x+6y-3x=3x-2-3x#

#rArr 5x+6y-3x=cancel(3x)-2-cancel(3x)#

#color(blue)(rArr 2x+6y=(-2)#

Hence, we can conclude that the equation

#color(red)(5x+6y = 3x-2#

is indeed a linear equation.

y-intercept can be found by setting #color(red)(x=0#

#rArr 2x+6y=(-2)#

#rArr 2(0)+6y=(-2)#

#rArr 6y= -2#

Divide both sides by #color(red)(6#

#rArr (6y)/6= (-2)/6#

#rArr y = -1/3#

Hence we understand that #color(blue)((0,-1/3)# is the y-intercept

To find the x-intercept, set #color(red)(y=0#

#rArr 2x+6y=(-2)#

#rArr 2x+6(0)=(-2)#

#rArr 2x=-2#

Divide both sides of the equation by #color(red)(2#

#rArr (2x)/2=(-2)/2#

#rArr (cancel 2x)/cancel 2=(-2)/2#

#rArr x = -1#

Hence we understand that #color(blue)((-1,0)# is the x-intercept

We can verify these results using a graph as given below:

I hope this explanation is helpful.