If the pair of equations has infinite solutions

Question

If the equations x + 9y = 12 and 3x + ky = m have infinite solutions, the value of (k + m) is -

A : 63
B : -63
C : 27
D : -27
E : Can not be determined

CAMANISHPARMAR · Accepted Answer

The two lines represented by x + 9y = 12 and 3x + ky = m will have infinite solutions if they are coincident lines, i.e. Two lines that lie on top of one another are called coincident lines. (Kindly refer to the attached graphical representation for more clarity). 6.png Line 1 - x + 9y = 12 . Multiply this equation of Line-1 with 3 to get 3x + 27y = 36 . Compare this with Line - 2 : 3x + ky = m . Hence k = 27 and m = 36. Therefore k + m = 63. (Ans) Kindly note that when you multiply this equation x + 9y = 12 by 3 you get a coincident line. Only coincident lines have infinite solutions.

KarishmaB · Answer

Two equations have infinite solutions when they represent the same line in the co-ordinate plane.

If the two equations are 
ax + by = c
dx + ey = f

a/d = b/e = c/f

The ratio of a/d is 1/3 so to get (k + m) we multiply (9+12)*3 = 63

generis · Answer

If two linear equations have infinite solutions, the equations are equivalent and describe the same line or "coincident lines."*

x + 9y = 12  (P) and
 3x + ky = m  (Q)

Make the equations equivalent

1) Start with the most restrictive condition: the x-coefficient in (Q)
The "3" in "3x" in (Q) is: 
-- not a variable; 
-- attached to a variable (unlike constant 12); and hence
-- the least changeable of our terms. 
For these equations to be equivalent, (P)'s x term  must be 3x  (or some multiple of 3 - worry about that later)

2) Multiply (P) by 3, write (Q) underneath:
 3x +  y = (36) 
 3x +  y = (m)

3) We can make the equations identical, such that solving will result in  0 = 0

Brackets and parentheses in #2 indicate terms that must be identical. Hence 
 k=27  and  m=36

(k+m)=(27+36)=63

Answer A

*Examples - these sets of lines have every point in common
3x + 4 = x + 2x + 1 + 3. Solve:
0 = 0. Always true. Infinite solutions. (Try plugging in a few numbers for x.)

3x + 2y = 10 (A)
6x + 4y = 20 (B) Multiply (A) by 2:
6x + 4y = 20 ... (A2). Subtract B from A2. 0 = 0

Heisenberg12 · Answer

If the equations x + 9y = 12 and 3x + ky = m have infinite solutions, the value of (k + m) is -

A : 63
B : -63
C : 27
D : -27
E : Can not be determined

In order for the above equations to have infinite solutions, they must represent the same line.

Thus, x+9y=12 and 3x+ky=m must be the same lines.

Multiplying equation x+9y=12 by 3, we get 3x+27y=36 -- equating this with 3x+ky=m, we get-

k=27, m=36.

Thus, k+m=27+36=63.

gracie · Answer

to get (k+m) alone on one side of equation, 
y must=-1 
thus, x=21
3*21+k*-1=m➡
k+m=63
A

prady2231 · Answer

clear explanation!! Adding to it and summering -

For lines - a1x + b1y = P
 a2x + b2y = Q

1: For lines to be parallel or No solution 
 a1/a2 = b1/b2 ≠ P/Q

2: For lines to be same or Infinite Solutions 
 a1/a2 = b1/b2 = P/Q

3 : For lines to be intersecting - 
 a1/a2 ≠ b1/b2

MathRevolution · Answer

For the infinite solution:  \frac{1}{3} = \frac{9}{k} = \frac{12}{m}

\frac{1}{3} = \frac{9}{k}  => k = 27

\frac{1}{3} = \frac{12}{m}  => m = 36

=> k + m = 27 + 36 = 63

Answer A

Fdambro294 · Answer

for the Equations to have an INFINITE Number of Solutions, the Equations must be EQUAL ---- therefore, an Infinite Number of Pairs of (X , Y) will Satisfy EACH Equation

x + 9y = 12

---multiply both sides of the equation by 3----

3x + 27y = 36

must be the SAME EQUATION as

3x + ky = m

-I- 27y = ky

and

-II- 36 = m

therefore, (k + m) = 27 + 36 = 63

-A-

luisdicampo · Answer

Deconstructing the Question  
We are given:
 x + 9y = 12 
 3x + ky = m 
The system has infinite solutions.

Step-by-step  
For infinite solutions, the second equation must be a multiple of the first.

Multiply the first equation by 3:
 3x + 27y = 36

Compare with:
 3x + ky = m

Thus:
 k = 27 
 m = 36

Compute:
 k + m = 27 + 36 = 63

Answer: 63 (A)

If the pair of equations has infinite solutions

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If the pair of equations has infinite solutions

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards