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

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.  

Multiple choice

Which of the following equations is a linear equation?

  1. 3x + 2yx2 = 4

  2. 2x + 3y2 = 4

  3. 2x + 2y + x2 = 7

  4. x = 4
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The highest power of the term in the given equation is called degree. An equation having degree one, is called a linear equation. $\because$x = 4 is an equation of degree one, therefore it is a linear equation.

Multiple choice

Which of the following equations is a linear equation?

  1. 7x + 10 = 0

  2. 2x + 3yz = 0

  3. 7xy + 2y = 0

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

The highest power of the term in the given equation is called degree. The equation having a degree one is called a linear equation. Therefore, 7x + 10 = 0 is the linear equation.

Multiple choice

Which of the following equations is a linear equation?

  1. xy + 2x2 = 3

  2. x + y - 3 = 0

  3. x2 + y2 - 3xy = 4

  4. x2 - y2 = 5
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The highest power of the term in the given equation is called degree. The equation having a degree one is called a linear equation. Therefore, the equation x + y - 3 = 0 is a linear equation.

Multiple choice statistics time series moving average and variation simple moving average

Find the equation of $y$ on $x$ on the basis of the following data:

$x$ $5$ $2$ $1$ $4$ $3$
$y$ $5$ $8$ $4$ $2$ $10$
  1. $y=-0.8x+7$
  2. $y=0.4x+7$
  3. $y=-0.4x+7$
  4. $y=-0.4x-7$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
 $x$  $y$  $xy$  $x^2$
 5  5  25  25
 2  8  16  4
 1  4  4  1
 4  2  8  16
 3  10  30  9
 $\sum x=15$  $\sum y=29$  $\sum xy=83$  $\sum x^2=55$

The linear equation be y=a+bx

$n$ is the number of observations
$n=5$
where $a=\dfrac{(\sum y)(\sum x^2)-(\sum x)(\sum xy)}{n(\sum x^2)-(\sum x)^2}$
$\implies a=\dfrac{(29)(55)-(15)(83)}{5(55)-(15)^2}=\dfrac{350}{50}=7$
and $b=\dfrac{n(\sum xy)-(\sum x)(\sum y)}{n(\sum x^2)-(\sum x)^2}$
$\implies b=\dfrac{(5)(83)-(15)(29)}{5(55)-(15)^2}=\dfrac{-20}{50}=-0.4$
Therefore, $y=-0.4x+7$
Multiple choice statistics time series moving average and variation simple moving average

Find the equation of $x$ on $y$ on the basis of the following data:

$x$ $2$ $4$ $6$ $8$ $10$
$y$ $6$ $5$ $4$ $3$ $2$
  1. $x+ 2y=14$
  2. $x-2y=14$
  3. $2x+ y=14$
  4. $2x- 2y=14$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the equation be $ax+by+c=0$


At $x=2 , y=6$
$\Rightarrow 2a+6b+c=0$

At $x=4, y=5$
$\Rightarrow 4a+5b+c=0$

At $x=6, y=4$
$\Rightarrow 6a+4b+c=0$

Solving these, we get
$a=1$
$b=$4
$c=-14$

$\therefore$ Equation is $x+2y=14$

Multiple choice statistics time series moving average and variation simple moving average

For the variables $x$ and $y$, the regression equations are given as $7x-3y-18=0$ and $4x-y-11=0$. Identify the regression equation of $y$ on $x$.

  1. $4x-y-11=0$
  2. $7x-3y-18=0$
  3. $4x-y-2=0$
  4. $7x-3y-4=0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let us assume that $7x-3y-18=0$ is the regression equation of $y$ on $x$.


Consider $7x-3y-18=0$

$\Rightarrow y=-6+\dfrac{7}{3}x$ 

$\therefore b_{yx}=\dfrac{7}{3}$

Now consider $4x-y-11=0$

$\Rightarrow x=\dfrac{11}{4}+\dfrac{1}{4}y$

$\therefore b_{xy}=\dfrac{1}{4}$

Now taking the product, $b_{yx} \times b_{xy}=\dfrac{7}{3} \times \dfrac{1}{4}=\dfrac{7}{12}<1$

Since the product is less than one, our assumptions are correct.

Thus $7x-3y-18=0$ is the regression equation of $y$ on $x$.

Multiple choice lines in planes applications of determinants inverse of a matrix and linear equations matrix algebra maths

$x + 2y + 3z = 0$ 

$2x + 3y + 4z = 0$
$7x + 13y + 19z =0$
The system of equations have non trivial solutions

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A system of homogeneous linear equations has non-trivial solutions if the determinant of the coefficient matrix is zero. The determinant of [[1, 2, 3], [2, 3, 4], [7, 13, 19]] is 1(57-52) - 2(38-28) + 3(26-21) = 5 - 20 + 15 = 0. Since the determinant is zero, non-trivial solutions exist.

Multiple choice lines in planes applications of determinants inverse of a matrix and linear equations matrix algebra maths

$2x - 3y + z = 0$ 

$x + 2y - 3z =0$
$4x - y - 2z = 0$
The system of equations have a non trivial solution

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For the system to have non-trivial solutions, the determinant must be zero. The determinant of [[2, -3, 1], [1, 2, -3], [4, -1, -2]] is 2(-4-3) - (-3)(-2+12) + 1(-1-8) = 2(-7) + 3(10) - 9 = -14 + 30 - 9 = 7. Since the determinant is not zero, the system has only the trivial solution (0,0,0).

Multiple choice lines in planes applications of determinants inverse of a matrix and linear equations matrix algebra maths

Which of the following are correct in respect of the system of equations $x + y + z = 8, x - y + 2z = 6$ and $3x - y + 5z = k$?
1. They have no solution, if $k = 15$.
2. They have infinitely many solutions, if $k = 20$.
3. They have unique solution, if $k = 25$.
Select the correct answer using the code given below

  1. 1 and 2 only

  2. 2 and 3 only

  3. 1 and 3 only

  4. 1, 2 and 3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

First we need to know if the above equations are linearly dependent or no,

In order to figure that out lets find the determinant $D$ which is;
$D=\left| \begin{matrix} 1 & 1 & 1 \ 1 & -1 & 2 \ 3 & -1 & 5 \end{matrix} \right| =0$
as $D=0$ the equations are linearly dependent, which means the system of equations can either be inconsistent or have infinitely many solutions. That shall depend on the value of k.
For there to be inifinitely many solutions , values of ${ D } _{ 1 }, { D } _{ 2 }, { D } _{ 3 }$ should be $0$ , defined as;
${ D } _{ 1 }=\left| \begin{matrix} 8 & 1 & 1 \ 6 & -1 & 2 \ k & -1 & 5 \end{matrix} \right| \ { D } _{ 2 }=\left| \begin{matrix} 1 & 8 & 1 \ 1 & 6 & 2 \ 3 & k & 5 \end{matrix} \right| \ { D } _{ 3 }=\left| \begin{matrix} 1 & 1 & 8 \ 1 & -1 & 6 \ 3 & -1 & k \end{matrix} \right| $
which on solving you get;
${ D } _{ 1 }=\left| \begin{matrix} 8 & 1 & 1 \ 6 & -1 & 2 \ k & -1 & 5 \end{matrix} \right| =3k-60=3(k-20)\ { D } _{ 2 }=\left| \begin{matrix} 1 & 8 & 1 \ 1 & 6 & 2 \ 3 & k & 5 \end{matrix} \right| =k-20\ { D } _{ 3 }=\left| \begin{matrix} 1 & 1 & 8 \ 1 & -1 & 6 \ 3 & -1 & k \end{matrix} \right| =40-2k=-2(k-20)$
Now for all the Determinants to be zero , its evident that $k=20$
Hence statement 2 is correct.
Subsequently for there to be no solution $k\neq 20$, hence statement 1 is correct as well,
As $D$ is zero the system of equations CANNOT have a unique solution hence, statement 3 is WRONG.