3.2 Systems of linear equations
4 min read•august 14, 2024
Systems of linear equations are a powerful tool for modeling real-world problems. They allow us to represent relationships between variables and find solutions that satisfy multiple conditions simultaneously. This topic builds on our understanding of linear equations and introduces methods for solving systems.
We'll explore various techniques for solving systems, including substitution, elimination, and graphing. We'll also discuss how to interpret solutions and apply these methods to practical scenarios, connecting mathematical concepts to real-world applications.
Solving systems of equations
Substitution method
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- Solve one equation for one variable
- Substitute the resulting expression into the other equation to solve for the remaining variable
- Most effective when one variable has a coefficient of 1 or -1 in one of the equations
- Example: Given the system and , solve the second equation for to get , then substitute this into the first equation to solve for
Elimination method
- Also known as the addition method
- Multiply one or both equations by a constant to eliminate one variable when the equations are added together
- Most effective when the coefficients of one variable are opposites or have a common factor
- If the resulting equation has no solution (0 = 1), the system has no solution
- If the resulting equation is an identity (0 = 0), the system has infinitely many solutions
- Example: Given the system and , multiply the first equation by -2 and add it to the second equation to eliminate the variable, then solve for
Number of solutions in systems
Types of solutions
- One solution (consistent and independent): Exactly one solution, represented by a single point where the lines intersect
- No solution (inconsistent): Represented by parallel lines that do not intersect
- Infinitely many solutions (consistent and dependent): Represented by coincident lines that overlap each other
Determining the number of solutions
- Graph the equations and identify the point(s) of intersection, if any
- Use the substitution or elimination methods to solve the system
- Compare the slopes and y-intercepts of the equations
- If the slopes are the same but the y-intercepts are different, the system has no solution (parallel lines)
- If the slopes and y-intercepts are the same, the system has infinitely many solutions (coincident lines)
- If the slopes are different, the system has one solution (intersecting lines)
Graphing systems of equations
Graphing linear equations
- Identify the slope and y-intercept of each equation
- Plot the y-intercept and use the slope to plot additional points
- Connect the points with a straight line
Interpreting the graph
- The point of intersection of the two lines represents the solution to the system
- If the lines are parallel (same slope but different y-intercepts), there is no point of intersection, and the system has no solution
- If the lines are coincident (same slope and same y-intercept), there are infinitely many points of intersection, and the system has infinitely many solutions
- Example: Given the system and , graph both equations on the same coordinate plane and identify the point of intersection (2, 5) as the solution
Applications of systems of equations
Modeling real-world problems
- Identify the unknown quantities and assign variables to represent them
- Create a system of linear equations by writing equations that represent the relationships between the unknown quantities based on the given information
- Solve the system using the , , or by graphing to find the values of the unknown quantities
- Interpret the solution in the context of the original problem, ensuring that the values make sense and answer the question being asked
- Verify the solution by substituting the values back into the original equations to ensure they satisfy both equations simultaneously
Examples of real-world applications
- Mixtures: Determining the quantities of different ingredients in a mixture given the total amount and the proportions of each ingredient
- Investments: Calculating the amounts invested at different interest rates to achieve a specific total return
- Rates: Finding the rates at which two objects travel given the distances covered and the time taken
- Example: A store sells two types of coffee beans, Premium and Regular. Premium beans cost 8 per pound. The store wants to create a 50-pound mixture that costs $9 per pound. How many pounds of each type of bean should be used in the mixture?
Key Terms to Review (16)
Augmented Matrix: An augmented matrix is a combination of the coefficients and constants from a system of linear equations, arranged in a rectangular format. This matrix representation allows for efficient manipulation and solution of the system using row operations. By representing the system in this way, one can easily apply techniques like Gaussian elimination or Gauss-Jordan elimination to find solutions or determine the nature of the solutions.
Coefficient Matrix: A coefficient matrix is a matrix that contains the coefficients of the variables in a system of linear equations. It is used to represent the system in a compact form, allowing for easier manipulation and solution using various methods like Gaussian elimination or matrix inversion. This matrix plays a crucial role in understanding the properties of the system, such as consistency and uniqueness of solutions.
Consistent System: A consistent system of equations is one where at least one solution exists that satisfies all equations simultaneously. This means that the equations do not contradict each other, allowing for the possibility of either a unique solution or infinitely many solutions. The graphical representation of such a system reveals lines or planes that intersect at least at one point, indicating the presence of solutions.
Dependent Equations: Dependent equations are systems of linear equations that have an infinite number of solutions, meaning that at least one equation can be derived from another. These equations represent the same line in a coordinate plane, illustrating that any solution of one equation is also a solution for the other. In practical terms, dependent equations suggest that there is a relationship between the variables where they do not intersect but rather overlap completely.
Determinant: A determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the matrix. It plays a crucial role in various mathematical concepts, such as solving systems of linear equations and understanding the behavior of linear transformations. The value of a determinant can help determine whether a system has a unique solution, no solution, or infinitely many solutions, and it can also indicate whether the matrix is invertible.
Elimination method: The elimination method is a technique used to solve systems of linear equations by eliminating one of the variables, allowing for the remaining variable to be solved more easily. This approach involves manipulating the equations to create a scenario where adding or subtracting them will remove one variable, simplifying the process of finding the solution. It is particularly useful when dealing with two or more equations simultaneously.
Graphing Lines: Graphing lines is the process of representing linear equations visually on a coordinate plane, where each point on the line corresponds to a solution of the equation. It provides a way to understand the relationship between two variables and how they change in relation to each other. This visual representation helps in solving systems of linear equations, identifying intersections, and understanding the behavior of linear functions.
Inconsistent System: An inconsistent system refers to a set of equations or inequalities that do not have any solutions, meaning there is no point that satisfies all the equations simultaneously. This occurs when the equations represent parallel lines in the case of linear equations, where they never intersect. Recognizing an inconsistent system is essential for understanding the relationships between equations and for determining whether a solution exists.
Independent Equations: Independent equations are linear equations in a system that have unique solutions and are not multiples of each other. This means that the lines represented by these equations will intersect at exactly one point in a two-dimensional space, indicating a single solution for the system. Independent equations are essential in understanding the behavior of systems of linear equations, as they ensure that each equation contributes distinct information to the solution set.
Infinite solutions: Infinite solutions occur when a system of linear equations has an unlimited number of solutions that satisfy all equations simultaneously. This situation arises when the equations represent the same line, indicating that every point on that line is a solution. It highlights a dependency among the equations, showcasing that they do not contradict each other but rather coincide perfectly.
Intersection point: An intersection point is the specific point where two or more lines, curves, or surfaces meet or cross each other in a coordinate plane. This point holds significant importance as it represents the solution to a system of linear equations, indicating where the equations share common values for their variables. Understanding intersection points is crucial when analyzing relationships between different mathematical models, as they reveal key insights about the system being studied.
Rank: Rank refers to the dimension of a matrix, which indicates the maximum number of linearly independent rows or columns. It provides insight into the solutions of systems of linear equations, determining whether the system has a unique solution, infinitely many solutions, or no solution at all. Understanding rank is essential for analyzing the behavior of linear systems and their associated matrices.
Slope-intercept form: Slope-intercept form is a way to express a linear equation in the format $$y = mx + b$$, where $$m$$ represents the slope of the line and $$b$$ represents the y-intercept. This form makes it easy to identify key features of the line, such as its steepness and where it crosses the y-axis. Understanding slope-intercept form is essential for solving linear equations and analyzing relationships between variables, especially when dealing with inequalities and systems of equations.
Solution Set: A solution set is a collection of all possible solutions that satisfy a given equation or inequality. It represents the values that make the equation true and can consist of a single number, multiple numbers, or even an entire range of values. Understanding the concept of a solution set is crucial for analyzing linear equations and inequalities, as well as systems of linear equations, since it provides insights into how these mathematical relationships function.
Standard Form: Standard form refers to a specific way of writing linear equations and inequalities, typically represented as $$Ax + By = C$$ for equations and $$Ax + By \leq C$$ or $$Ax + By \geq C$$ for inequalities, where A, B, and C are integers and A and B are not both zero. This format helps in easily identifying the slope and intercepts of a line, making it straightforward to work with linear relationships and facilitate solutions in various mathematical contexts.
Substitution Method: The substitution method is a technique used to solve systems of equations by expressing one variable in terms of another and then substituting that expression into a different equation. This approach simplifies the process of finding the values of the variables by allowing one equation to be reduced to a single variable, making it easier to solve. It's particularly useful in linear equations, where two or more equations need to be solved simultaneously, as well as in systems involving differential equations.