Challenge: Solve a System of Two Linear Equations
A system of linear equations consists of two or more equations that share the same set of variables. In the case of two equations with two variables, each equation describes a line on a coordinate plane. The solution to the system is the point where these two lines intersect, representing the values of the variables that satisfy both equations simultaneously.
Systems of linear equations are commonly used in real-world scenarios such as:
- Finding the intersection point of two roads;
- Determining how two budget constraints overlap;
- Solving for unknowns in mixture and investment problems.
By solving these systems, you can answer questions like: At what price and quantity do supply and demand curves meet? or How do two different payment plans compare over time?
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Write a function that solves a system of two linear equations with two variables using the coefficients and constants from each equation. Your function should determine whether a unique solution exists and, if so, return it as a tuple (x, y). If no unique solution exists, the function should raise an exception.
Follow these steps:
- Define a function that takes six arguments: the coefficients and constants for two equations (
a1,b1,c1,a2,b2,c2). - The equations are in the form:
a1 * x + b1 * y = c1a2 * x + b2 * y = c2
- Calculate the determinant of the coefficient matrix:
det = a1 * b2 - a2 * b1. - If the determinant is zero, raise an exception to indicate that there is no unique solution for the system.
- If the determinant is not zero:
- Compute the value of
xusing the formula:(c1 * b2 - c2 * b1) / det. - Compute the value of
yusing the formula:(a1 * c2 - a2 * c1) / det. - Return the solution as a tuple
(x, y).
- Compute the value of
- Make sure your function uses the given coefficients and constants to find the solution and only returns the unique solution if it exists.
Oplossing
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Can you explain the different methods for solving systems of linear equations?
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A system of linear equations consists of two or more equations that share the same set of variables. In the case of two equations with two variables, each equation describes a line on a coordinate plane. The solution to the system is the point where these two lines intersect, representing the values of the variables that satisfy both equations simultaneously.
Systems of linear equations are commonly used in real-world scenarios such as:
- Finding the intersection point of two roads;
- Determining how two budget constraints overlap;
- Solving for unknowns in mixture and investment problems.
By solving these systems, you can answer questions like: At what price and quantity do supply and demand curves meet? or How do two different payment plans compare over time?
Swipe to start coding
Write a function that solves a system of two linear equations with two variables using the coefficients and constants from each equation. Your function should determine whether a unique solution exists and, if so, return it as a tuple (x, y). If no unique solution exists, the function should raise an exception.
Follow these steps:
- Define a function that takes six arguments: the coefficients and constants for two equations (
a1,b1,c1,a2,b2,c2). - The equations are in the form:
a1 * x + b1 * y = c1a2 * x + b2 * y = c2
- Calculate the determinant of the coefficient matrix:
det = a1 * b2 - a2 * b1. - If the determinant is zero, raise an exception to indicate that there is no unique solution for the system.
- If the determinant is not zero:
- Compute the value of
xusing the formula:(c1 * b2 - c2 * b1) / det. - Compute the value of
yusing the formula:(a1 * c2 - a2 * c1) / det. - Return the solution as a tuple
(x, y).
- Compute the value of
- Make sure your function uses the given coefficients and constants to find the solution and only returns the unique solution if it exists.
Oplossing
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