Solving linear systems (solve) in SciPy

Introduction

We use solving linear systems to find unknown values that satisfy multiple equations at once. It helps us solve real problems like balancing budgets or mixing ingredients.

When you want to find the values of variables that satisfy several equations together.

When you have data relationships expressed as linear equations and need exact solutions.

When modeling real-world problems like electrical circuits or resource allocation.

When you want to solve equations quickly using computer programs.

Syntax

SciPy

from scipy.linalg import solve

x = solve(A, b)

A is a square matrix representing coefficients of the equations.

b is the right-hand side vector or matrix of constants.

Examples

Solves two equations with two unknowns and prints the solution.

SciPy

from scipy.linalg import solve

A = [[3, 1], [1, 2]]
b = [9, 8]
x = solve(A, b)
print(x)

Solves three equations with three unknowns.

SciPy

from scipy.linalg import solve

A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]
b = [14, 13, 23]
x = solve(A, b)
print(x)

Sample Program

This program solves two linear equations with two unknowns and prints the values of x and y.

SciPy

from scipy.linalg import solve

# Coefficients matrix for equations:
# 2x + 3y = 8
# 5x + 4y = 13
A = [[2, 3], [5, 4]]

# Constants on right side
b = [8, 13]

# Solve for x and y
solution = solve(A, b)

print(f"Solution for x and y: {solution}")

Important Notes

The matrix A must be square (same number of rows and columns).

If A is singular (no unique solution), solve will raise an error.

Use numpy arrays or lists for A and b; scipy will convert them internally.

Summary

Use scipy.linalg.solve to find exact solutions to linear equations.

Provide the coefficient matrix and constants vector to get the unknowns.

This method is fast and reliable for well-formed linear systems.

Practice

1. What does the scipy.linalg.solve function do in solving linear systems?

easy

A. It finds the exact solution vector for a system of linear equations.

B. It approximates the solution using random guesses.

C. It only checks if the system has any solution without solving.

D. It multiplies two matrices together.

Solution

1. Step 1: Understand the purpose of scipy.linalg.solve

   This function is designed to solve linear systems of the form Ax = b, where A is a matrix and b is a vector.

2. Step 2: Identify the output of the function

   The function returns the vector x that exactly satisfies the equation Ax = b.

3. Final Answer:

   It finds the exact solution vector for a system of linear equations. -> Option A

4. Quick Check:

   Function purpose = exact solution vector [OK]

Hint: Remember: solve() returns exact solutions, not guesses. [OK]

Common Mistakes:

• Thinking solve() approximates solutions
• Confusing solve() with matrix multiplication
• Assuming solve() only checks for solution existence

2. Which of the following is the correct way to import the solve function from scipy.linalg?

easy

A. from scipy.linalg import solve

B. import solve from scipy.linalg

C. from scipy import solve.linalg

D. import scipy.solve.linalg

Solution

1. Step 1: Recall Python import syntax for functions

   To import a specific function from a module, use from module import function.

2. Step 2: Match the correct module and function

   The function solve is inside the scipy.linalg module, so the correct import is from scipy.linalg import solve.

3. Final Answer:

   from scipy.linalg import solve -> Option A

4. Quick Check:

   Correct import syntax = from scipy.linalg import solve [OK]

Hint: Use 'from module import function' for specific imports. [OK]

Common Mistakes:

• Using 'import solve from ...' which is invalid syntax
• Trying to import submodules as functions
• Using dot notation incorrectly in import statements

3. What is the output of this code?

import numpy as np
from scipy.linalg import solve
A = np.array([[2, 1], [1, 3]])
b = np.array([10, 15])
x = solve(A, b)
print(x)

medium

A. [4. 3.]

B. [3. 4.]

C. [2. 5.]

D. [5. 2.]

Solution

1. Step 1: Set up the system of equations

   The matrix A and vector b represent the system:
   2x + y = 10
   x + 3y = 15

2. Step 2: Solve the system manually

   Solving manually:
   Multiply first equation by 3: 6x + 3y = 30
   Subtract second equation: 5x = 15 -> x = 3
   y = 10 - 2*3 = 4
   Thus, x = [3. 4.], which matches the output of solve(A, b).

3. Final Answer:

   [3. 4.] -> Option B

4. Quick Check:

   solve(A,b) output = [3. 4.] [OK]

Hint: Trust solve() output for exact numeric solutions. [OK]

Common Mistakes:

• Mixing up order of solution vector
• Trying to solve manually without decimals
• Assuming integer solutions only

4. What is wrong with this code snippet?

import numpy as np
from scipy.linalg import solve
A = np.array([[1, 2], [2, 4]])
b = np.array([5, 10])
x = solve(A, b)
print(x)

medium

A. The code will run and print the solution without issues.

B. Vector b has wrong shape and causes a syntax error.

C. The import statement is incorrect.

D. Matrix A is singular (not invertible), so solve() will raise an error.

Solution

1. Step 1: Check matrix A properties

   Matrix A = [[1, 2], [2, 4]] has rows that are multiples (second row = 2 * first row), so it is singular (determinant = 0).

2. Step 2: Understand solve() behavior with singular matrices

   Since A is singular, it cannot be inverted, so solve() will raise a LinAlgError indicating the matrix is singular.

3. Final Answer:

   Matrix A is singular (not invertible), so solve() will raise an error. -> Option D

4. Quick Check:

   Singular matrix causes error in solve() [OK]

Hint: Check if matrix rows are multiples before solving. [OK]

Common Mistakes:

• Ignoring singular matrix errors
• Assuming solve() works for all matrices
• Confusing vector shape errors with matrix singularity

5. You have a system of 3 equations with 3 unknowns:

A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]
b = [11, 7, 28]

Using scipy.linalg.solve, what is the solution vector x?

hard

A. [1. 2. 3.]

B. [0. 1. 2.]

C. [2. 3. 1.]

D. [3. 1. 2.]

Solution

1. Step 1: Write the system of equations

   The system is:
   1x + 2y + 3z = 11
   0x + 1y + 4z = 7
   5x + 6y + 0z = 28

2. Step 2: Solve the system using substitution

   From second equation: y = 7 - 4z
   Substitute into first: x + 2(7 - 4z) + 3z = 11 -> x + 14 - 8z + 3z = 11 -> x - 5z = -3 -> x = -3 + 5z
   Substitute into third: 5(-3 + 5z) + 6(7 - 4z) = 28 -> -15 + 25z + 42 - 24z = 28 -> z + 27 = 28 -> z = 1
   y = 7 - 4*1 = 3
   x = -3 + 5*1 = 2
   Thus, x = [2. 3. 1.]

3. Final Answer:

   [2. 3. 1.] -> Option C

4. Quick Check:

   solve(A,b) = [2. 3. 1.] [OK]

Hint: Use solve() to avoid manual errors in 3x3 systems. [OK]

Common Mistakes:

• Manual calculation errors with substitution
• Mixing variable order in solution vector
• Ignoring code output for complex systems