SOLUTION: determine a and b for which the system of linear equation has infinite no. of solutions: (2a-1) + 3y -5 =0 3x + (b-1)y -2=0

Question 905340: determine a and b for which the system of linear equation has infinite no. of solutions: (2a-1) + 3y -5 =0
3x + (b-1)y -2=0
Found 2 solutions by mananth, Theo:
Answer by mananth(16946) (Show Source): You can put this solution on YOUR website!
determine a and b for which the system of linear equation has infinite no. of solutions: (2a-1)x + 3y -5 =0
3x + (b-1)y -2=0
in the linear system of equations
a1x+b1y=c1
a2x+b2y=c2
For the lines have infinite number of solutions

here a1= (2a-1) ,b1=3, c1= 5
a2=3,b2=(b-1), c2=2

2(2a-1)=15
4a-2=15
4a=17
a=17/4
3/(b-1) =5/2
6=5(b-1)
6=5b-5
5b=11
b=11/5
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
you will get an infinite number of solutions when the two equations completely cancel each other out when you subtract one equation from the other.

The result will be 0 = 0 which is a true statement indicating that the equations are identical and will completely overlap on the graph resulting in what looks like just one line when in reality you are graphing 2 lines.

after i go through it, you will see what i mean.

your 2 equations are:

(2a-1)x + 3y = 5
3x + (b-1)y = 2

in order for the equations to be identical, the constants will have to be identical.

you can get that to happen by multiplying both sides of the first equation by 2 and multiplying both sides of the second equation by 5.

you will get:

2*(2a-1)x + 2*3y = 2*5
5*3x + 5*(b-1)y = 5*2

simplify these equations and you will get:

(4a-2)x + 6y = 10
15x + (5b-5)y = 10

for these equations to be identical:

(4a-2) must be equal to 15 and (5b-5) must be equal to 6

solve for a to get a = 17/4
solve for b to get b = 11/5

i'm assuming you know how to solve for a and b.
if you don't, let me know and i'll guide you through it.

replace a with 17/4 and replace b with 11/5 in your original equations.

your original equations are:

(2a-1)x + 3y = 5
3x + (b-1)y = 2

replace a with 17/4 and replace b with 11/5 and you will get:

(30/4)x + 3y = 5
3x + (6/5)y = 2

convert each of these equations to slope intercept form.

(30/4)x + 3y = 5 becomes y = -(5/2)x + (5/3)
3x + (6/5)y = 2 becomes y = -(5/2)x + (5/3)

i'm assuming you know how to convert from standard form of ax+by=c to slope intercept form of y=mx+b.
if you don't, let me know and i'll guide you through it.

the two equations in slope intercept form are identical to each other which means they will be the same line when they are graphed which means that every point on the first line will also be on the second line which means they will have an infinite number of simultaneous solutions.

if the slope is the same and the y-intercept is the same, the lines will be identical and you will have an infinite number of solutions.

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