subract ax+by+c from 2by+3c100 points ka gsi loot lo
Answers
Step-by-step explanation:
Basic Equations of Lines and Planes
Equation of a Line
An important topic of high school algebra is "the equation of a line." This means an equation in x and y whose solution set is a line in the (x,y) plane.
The most popular form in algebra is the "slope-intercept" form
y = mx + b.
This in effect uses x as a parameter and writes y as a function of x: y = f(x) = mx+b. When x = 0, y = b and the point (0,b) is the intersection of the line with the y-axis.
Thinking of a line as a geometrical object and not the graph of a function, it makes sense to treat x and y more evenhandedly. The general equation for a line (normal form) is
ax + by = c,
with the stipulation that at least one of a or b is nonzero. This can easily be converted to slope-intercept form by solving for y:
y = (-a/b) + c/b,
except for the special case b = 0, when the line is parallel to the y-axis.
If the coefficients on the normal form are multiplied by a nonzero constant, the set of solutions is exactly the same, so, for example, all these equations have the same line as solution.
2x + 3 y = 4
4x + 6y = 8
-x - (3/2) y = -2
(1/2)x + (3/4)y = 1
In general, if k is a nonzero constant, then these are equations for the same line, since they have the same solutions.
ax + by = c
(ka)x + (kb)y = kc.
Answer:
Given equations are,
ax+by=c ....(1)
and bx+ay=1+c ....(2)
Multiply equation (1) by a and equation (2) by b, we get
a
2
x+aby=ac ....(3)
and b
2
x+aby=b+bc ....(4)
Subtract equations (3) and (4),
x(a
2
−b
2
)=ac−b−bc
⇒x=
a
2
−b
2
ac−b−bc
Put this value in equation (1), we get
a(
a
2
−b
2
ac−b−bc
)+by=c
⇒a
2
c−ab−abc−a
2
c+b
2
c=−by(a
2
−b
2
)
⇒b(bc−ab−a)=−by(a
2
−b
2
)
⇒y=
b
2
−a
2
bc−a−ac