For what value of k pair of linear equations have unique solution?

The GMAT sample question in quant given below is a Linear Equations question and tests concepts related to types of solutions for a system of linear equations. This concept is usually tested in the GMAT as a data sufficiency question rather than as a problem solving question. A sub 600 level GMAT practice question in system of linear equations.

Question 3: For what values of 'k' will the pair of equations 3x + 4y = 12 and kx + 12y = 30 NOT have a unique solution?

1. 9
2. 12
3. 3
4. 7.5
5. 2.5

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Explanatory Answer

Condition for Unique Solution to Linear Equations

A system of linear equations ax + by + c = 0 and dx + ey + g = 0 will have a unique solution if the two lines represented by the equations ax + by + c = 0 and dx + ey + g = 0 intersect at a point.
i.e., if the two lines are neither parallel nor coincident.
Essentially, the slopes of the two lines should be different.

What does that translate into?

ax + by + c = 0 and dx + ey + g = 0 will intersect at one point if their slopes are different.
Express both the equations in the standardized y = mx + c format, where 'm' is the slope of the line and 'c' is the y-intercept.

ax + by + c = 0 can be written as y = \-\frac{a}{b}x -\frac{c}{a} )
And dx + ey + g = 0 can be written as y = \-\frac{d}{e}x -\frac{g}{e} )
Slope of the first line is \-\frac{a}{b} ) and that of the second line is \-\frac{d}{e} )
For a unique solution, the slopes of the lines should be different.
∴ \-\frac{a}{b} \neq -\frac{d}{e} )
Or \\frac{a}{d} \neq \frac{b}{e} )

Condition for the equations to NOT have a unique solution

The slopes should be equal
Or \\frac{a}{d} = \frac{b}{e} )

Apply the condition in the given equations to find k

In the question given above, a = 3, b = 4, d = k and e = 12.
Therefore, \\frac{3}{k} = \frac{4}{12} )
Or 'k' should be equal to 9 for the system of linear equations to NOT have a unique solution.

The question is "What is the value of k?
When k = 9, the system of equations will represent a pair of parallel lines (their y-intercepts are different). So, there will be NO solution to this system of linear equations in two variables.

Choice A is the correct answer.

For what value of k, the system of equations
x + 2y = 3,
5x + ky + 7 = 0
Have (i) a unique solution, (ii) no solution?
Also, show that there is no value of k for which the given system of equation has infinitely namely solutions

The given system of equations:
x + 2y = 3
⇒ x + 2y - 3 = 0                           ….(i)
And, 5x + ky + 7 = 0                   …(ii)
These equations are of the following form:
`a_1x+b_1y+c_1 = 0, a_2x+b_2y+c_2 = 0`
where, `a_1 = 1, b_1= 2, c_1= -3 and a_2 = 5, b_2 = k, c_2 = 7`
(i) For a unique solution, we must have:
∴ `(a_1)/(a_2) ≠ (b_1)/(b_2) i.e., 1/5 ≠ 2/k ⇒ k ≠ 10`
Thus for all real values of k other than 10, the given system of equations will have a unique solution.
(ii) In order that the given system of equations has no solution, we must have:
`(a_1)/(a_2) = (b_1)/(b_2 )≠ (c_1)/(c_2)`
`⇒ 1/5 ≠ 2/k ≠ (−3)/7`
`⇒ 1/5 ≠ 2/k and 2/k ≠ (−3)/7`
`⇒k = 10, k ≠ 14/(−3)`
Hence, the required value of k is 10.
There is no value of k for which the given system of equations has an infinite number of solutions.

While DonAntonio's answer is certainly correct, it is likely that your question comes from a class where determinants have not yet been discussed, so you may need a different perspective.

In that case, recall that your system will be inconsistent if, after row reduction, you have a row of the form $( 0 \ 0 \ 0 \mid 1)$ since this row would correspond to the equation $0x+0y+0z=1$ which clearly has no solutions.

On the other hand, you also don't want a row of of the form $( 0 \ 0 \ 0 \mid 0)$, which would give you a free variable and hence infinitely many solutions.

Thus, you should find the values of $k$ for which $2-6k-4k^2 = 0$. By our discussion, we can see that as long as you avoid those values of $k$, your system will have a solution, and this solution will be unique.

You can find these "bad" values of $k$ by methods from high school algebra, e.g. the quadratic formula.

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