Mathematics · Quantitative Aptitude

Linear Equations

153 Questions

Linear equations involve solving for unknown variables in single or multi variable systems. These questions test algebraic manipulation and logical consistency skills. They are a core component of quantitative aptitude and advanced mathematics tests.

Solving simultaneous equationsSingle variable equationsSystem consistency checksIndeterminate equationsMatrix form solutions

Linear Equations Questions

Multiple choice

The linear equation 2x + 3y - 5 = 0 has

  1. no solution

  2. exactly one solution

  3. two solutions

  4. three solutions

  5. infinitely many solutions
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

Each solution (x, y) of this linear equation corresponds to a point on a line. Since a line consists of infinitely large number of points, the linear equation has infinitely many solutions.

Multiple choice

For what values of $\alpha$ and $\beta$ will the following simultaneous equations have an infinite number of solutions? x + y + z = 5; x + 3y + 3z = 9; x + 2y + $\alpha Z$ = $\beta$

  1. 2, 7

  2. 3, 8

  3. 8, 3

  4. 7, 2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For infinite solutions, the augmented matrix must have a rank less than the number of variables. Solving the system x+y+z=5 and x+3y+3z=9 gives 2y+2z=4, or y+z=2. Substituting into the third equation x+2y+alpha*z=beta, we need the equations to be consistent, which occurs when alpha=2 and beta=7.

Multiple choice

In the solution of the following set of linear equations by Gauss elimination using partial pivoting 5x + y + 2z = 34; 4y − 3z = 12 and 10x − 2y + z = −4 The pivots for elimination of x and y are

  1. 10 and 4

  2. 10 and 2

  3. 5 and 4

  4. 5 and – 4
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Partial pivoting requires choosing the largest absolute value in the current column as the pivot. For the first column (x), the coefficients are 5, 0, 10; the largest is 10. After eliminating x, the remaining system is processed, and the largest available coefficient for y is chosen.

Multiple choice

x + 2y + z = 4
2x + y + 2z = 5 x - y + z = 1

The system of algebraic equations given above has

  1. a unique solution of x = 1, y = 1 and z = 1

  2. only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0)

  3. infinite number of solutions

  4. no feasible solution
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\text{For given equation matrix form is as follow}\\ A = \begin{bmatrix} \ 1 & 2 & 1 \ \ 2 & 1 & 2 \ \ 1 & -1 & 1 \ \end{bmatrix}, B = \begin{bmatrix} \ 4 \ \ 5 \ \ 1 \ \end{bmatrix}$

Multiple choice

For what value(s) of k does the given system of equations have no solution?

kx + 2y = 3 3x + 6y = 10

  1. k = 3/10

  2. k = 1

  3. Either k = 3/10 or k = 1

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

kx + 2y = 3 3x + 6y = 10 Comparing above equations with  a1x + b1y = c1 a2x + b2y = c2 We get $\frac{a_1}{a_2} = \frac{k}{3}$ , $\frac{b_1}{b_2} = \frac{2}{6}$ , $\frac{c_1}{c_2} = \frac{3}{10}$ For no solution: a1/a2  = b1/b2 $ \neq$ c1/c2 $\Rightarrow$ k/3  = 2/6 $ \neq$ 3/10 $\Rightarrow$ k = 1 $ \neq$ 9/10 The given system of equations will have no solution if k  =  1.