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

Solve the system of equations: (2x + 3y = 7) and (4x - y = 5).

  1. \((x, y) = (1, 2)\)
  2. \((x, y) = (2, 1)\)
  3. \((x, y) = (3, 0)\)
  4. \((x, y) = (0, 3)\)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

To solve the system of equations, we can use the substitution method. First, solve one of the equations for one of the variables. For example, we can solve the first equation for (x): (x = (7 - 3y)/2). Then, substitute this expression for (x) into the other equation: (4((7 - 3y)/2) - y = 5). This gives us (14 - 6y - y = 5), which simplifies to (-7y = -9). Dividing both sides by (-7), we get (y = 9/7). Substituting this value of (y) back into the first equation, we get (2x + 3(9/7) = 7), which simplifies to (x = 2). Therefore, the solution to the system of equations is ((x, y) = (2, 1)).

Multiple choice

Solve the system of equations: (x + 2y = 4) and (3x - y = 1).

  1. \((x, y) = (1, 1)\)
  2. \((x, y) = (2, 0)\)
  3. \((x, y) = (3, -1)\)
  4. \((x, y) = (4, -2)\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To solve the system of equations, we can use the elimination method. First, multiply the second equation by 2 to get (6x - 2y = 2). Then, add this equation to the first equation to get (7x = 6). Dividing both sides by 7, we get (x = 6/7). Substituting this value of (x) back into the first equation, we get ((6/7) + 2y = 4), which simplifies to (y = 1). Therefore, the solution to the system of equations is ((x, y) = (1, 1)).

Multiple choice

Solve the system of equations: (2x + y = 5) and (x - 2y = 3).

  1. \((x, y) = (2, 1)\)
  2. \((x, y) = (3, 0)\)
  3. \((x, y) = (4, -1)\)
  4. \((x, y) = (5, -2)\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To solve the system of equations, we can use the substitution method. First, solve one of the equations for one of the variables. For example, we can solve the first equation for (x): (x = (5 - y)/2). Then, substitute this expression for (x) into the other equation: ((5 - y)/2 - 2y = 3). This gives us (5 - y - 4y = 6), which simplifies to (-5y = 1). Dividing both sides by (-5), we get (y = -1/5). Substituting this value of (y) back into the first equation, we get (2x + (-1/5) = 5), which simplifies to (x = 2). Therefore, the solution to the system of equations is ((x, y) = (2, 1)).

Multiple choice

Solve the system of equations: (3x - 2y = 1) and (2x + y = 4).

  1. \((x, y) = (1, 2)\)
  2. \((x, y) = (2, 1)\)
  3. \((x, y) = (3, 0)\)
  4. \((x, y) = (0, 3)\)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

To solve the system of equations, we can use the substitution method. First, solve one of the equations for one of the variables. For example, we can solve the second equation for (y): (y = 4 - 2x). Then, substitute this expression for (y) into the other equation: (3x - 2(4 - 2x) = 1). This gives us (3x - 8 + 4x = 1), which simplifies to (7x = 9). Dividing both sides by 7, we get (x = 9/7). Substituting this value of (x) back into the second equation, we get (2(9/7) + y = 4), which simplifies to (y = 1). Therefore, the solution to the system of equations is ((x, y) = (2, 1)).

Multiple choice

Solve the system of equations: (4x + 3y = 11) and (2x - y = 1).

  1. \((x, y) = (2, 3)\)
  2. \((x, y) = (3, 2)\)
  3. \((x, y) = (4, 1)\)
  4. \((x, y) = (1, 4)\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To solve the system of equations, we can use the elimination method. First, multiply the second equation by 3 to get (6x - 3y = 3). Then, add this equation to the first equation to get (10x = 14). Dividing both sides by 10, we get (x = 14/10 = 7/5). Substituting this value of (x) back into the first equation, we get (4(7/5) + 3y = 11), which simplifies to (y = 3). Therefore, the solution to the system of equations is ((x, y) = (2, 3)).

Multiple choice

Solve the system of equations: (x - y = 2) and (2x + y = 7).

  1. \((x, y) = (3, 1)\)
  2. \((x, y) = (4, 2)\)
  3. \((x, y) = (5, 3)\)
  4. \((x, y) = (6, 4)\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To solve the system of equations, we can use the addition method. First, add the two equations together to get (3x = 9). Dividing both sides by 3, we get (x = 3). Substituting this value of (x) back into the first equation, we get (3 - y = 2), which simplifies to (y = 1). Therefore, the solution to the system of equations is ((x, y) = (3, 1)).

Multiple choice

Solve the system of equations: (2x + 3y = 8) and (3x - 2y = 1).

  1. \((x, y) = (1, 2)\)
  2. \((x, y) = (2, 3)\)
  3. \((x, y) = (3, 4)\)
  4. \((x, y) = (4, 5)\)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

To solve the system of equations, we can use the elimination method. First, multiply the second equation by 2 to get (6x - 4y = 2). Then, add this equation to the first equation to get (8x - y = 10). Dividing both sides by 8, we get (x - (1/8)y = (5/4)). Now, multiply the second equation by 3 to get (9x - 6y = 3). Then, add this equation to the first equation to get (17x - 7y = 8). Dividing both sides by 17, we get (x - (7/17)y = (8/17)). Now, we have two equations in two variables: (x - (1/8)y = (5/4)) and (x - (7/17)y = (8/17)). We can solve this system of equations using the substitution method. First, solve one of the equations for (x). For example, we can solve the first equation for (x): (x = (5/4) + (1/8)y). Then, substitute this expression for (x) into the other equation: ((5/4) + (1/8)y - (7/17)y = (8/17)). This gives us (17(5/4) + 17(1/8)y - 17(7/17)y = 17(8/17)), which simplifies to (y = 3). Substituting this value of (y) back into the first equation, we get (x - (1/8)(3) = (5/4)), which simplifies to (x = 2). Therefore, the solution to the system of equations is ((x, y) = (2, 3)).

Multiple choice

Solve the system of equations: (x + 2y = 5) and (2x - y = 3).

  1. \((x, y) = (1, 2)\)
  2. \((x, y) = (2, 1)\)
  3. \((x, y) = (3, 0)\)
  4. \((x, y) = (0, 3)\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To solve the system of equations, we can use the substitution method. First, solve one of the equations for one of the variables. For example, we can solve the first equation for (x): (x = 5 - 2y). Then, substitute this expression for (x) into the other equation: (2(5 - 2y) - y = 3). This gives us (10 - 4y - y = 3), which simplifies to (-5y = -7). Dividing both sides by (-5), we get (y = 7/5). Substituting this value of (y) back into the first equation, we get (x + 2(7/5) = 5), which simplifies to (x = 1). Therefore, the solution to the system of equations is ((x, y) = (1, 2)).

Multiple choice

Solve the system of equations: (3x - 2y = 7) and (2x + y = 4).

  1. \((x, y) = (1, 2)\)
  2. \((x, y) = (2, 3)\)
  3. \((x, y) = (3, 4)\)
  4. \((x, y) = (4, 5)\)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To solve the system of equations, we can use the elimination method. First, multiply the second equation by 2 to get (4x + 2y = 8). Then, add this equation to the first equation to get (7x = 15). Dividing both sides by 7, we get (x = 15/7). Substituting this value of (x) back into the second equation, we get (2(15/7) + y = 4), which simplifies to (y = 4). Therefore, the solution to the system of equations is ((x, y) = (3, 4)).

Multiple choice

Solve the system of equations: (x + y = 5) and (x - y = 1).

  1. \((x, y) = (2, 3)\)
  2. \((x, y) = (3, 2)\)
  3. \((x, y) = (4, 1)\)
  4. \((x, y) = (1, 4)\)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

To solve the system of equations, we can use the addition method. First, add the two equations together to get (2x = 6). Dividing both sides by 2, we get (x = 3). Substituting this value of (x) back into the first equation, we get (3 + y = 5), which simplifies to (y = 2). Therefore, the solution to the system of equations is ((x, y) = (3, 2)).

Multiple choice

Solve the system of equations: (2x + 3y = 7) and (4x - y = 5).

  1. \((x, y) = (1, 2)\)
  2. \((x, y) = (2, 1)\)
  3. \((x, y) = (3, 0)\)
  4. \((x, y) = (0, 3)\)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

To solve the system of equations, we can use the substitution method. First, solve one of the equations for one of the variables. For example, we can solve the first equation for (x): (x = (7 - 3y)/2). Then, substitute this expression for (x) into the other equation: (4((7 - 3y)/2) - y = 5). This gives us (14 - 6y - y = 5), which simplifies to (-7y = -9). Dividing both sides by (-7), we get (y = 9/7). Substituting this value of (y) back into the first equation, we get (2x + 3(9/7) = 7), which simplifies to (x = 2). Therefore, the solution to the system of equations is ((x, y) = (2, 1)).

Multiple choice

What is the solution to the equation x^2 - 4x + 3 = 0?

  1. x = 1, x = 3

  2. x = -1, x = -3

  3. x = 2, x = 3

  4. x = -2, x = -3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We can solve this equation using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. Substituting the values of a, b, and c, we get x = (-(-4) ± √((-4)^2 - 4(1)(3))) / 2(1). Simplifying this equation, we get x = (4 ± √(16 - 12)) / 2. Further simplifying, we get x = (4 ± √4) / 2. Therefore, the solutions to the equation are x = 1 and x = 3.

Multiple choice

What is the solution to the system of linear equations (\begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}\begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix})?

  1. \(x = 1, y = 2, z = 3\)
  2. \(x = 2, y = 3, z = 4\)
  3. \(x = 3, y = 4, z = 5\)
  4. \(x = 4, y = 5, z = 6\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The solution to the system of linear equations (\begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}\begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}) is (x = 1, y = 2, z = 3).

Multiple choice

What is the solution to the system of linear equations (\begin{bmatrix} 2 & 1 & 1 \ 4 & 3 & 3 \ 6 & 5 & 5 \end{bmatrix}\begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix})?

  1. \(x = 1, y = 2, z = 3\)
  2. \(x = 2, y = 3, z = 4\)
  3. \(x = 3, y = 4, z = 5\)
  4. \(x = 4, y = 5, z = 6\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The solution to the system of linear equations (\begin{bmatrix} 2 & 1 & 1 \ 4 & 3 & 3 \ 6 & 5 & 5 \end{bmatrix}\begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}) is (x = 1, y = 2, z = 3).

Multiple choice

What is the solution to the system of linear equations (\begin{bmatrix} 1 & 2 & 3 \ 2 & 5 & 4 \ 3 & 1 & 2 \end{bmatrix}\begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix})?

  1. \(x = 1, y = 2, z = 3\)
  2. \(x = 2, y = 3, z = 4\)
  3. \(x = 3, y = 4, z = 5\)
  4. \(x = 4, y = 5, z = 6\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The solution to the system of linear equations (\begin{bmatrix} 1 & 2 & 3 \ 2 & 5 & 4 \ 3 & 1 & 2 \end{bmatrix}\begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}) is (x = 1, y = 2, z = 3).