Mathematics · Quantitative Aptitude

Linear Equations

196 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 pair of linear equations x = 4 and x = 6 gives

  1. no solution

  2. one solution

  3. two solutions

  4. three solutions

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

The two equations represent parallel lines. They do not intersect and hence, give no common solution.

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

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