5.9 Systems of Linear Equations in Two Variables
3 min read•june 18, 2024
Linear equations are powerful tools for solving real-world problems. They help us model relationships between variables and find solutions to complex situations. Understanding how to work with these equations is crucial for tackling everyday challenges.
Systems of linear equations take problem-solving to the next level. By combining multiple equations, we can solve for multiple unknowns simultaneously. This approach is invaluable in fields like economics, engineering, and science.
Solving Systems of Linear Equations
Methods for solving linear systems
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- involves plotting both equations on the same
- The point where the two lines intersect represents the solution to the system
- If the lines are parallel (same but different y-intercepts), the system has
- If the lines are coincident (same slope and ), the system has
- involves solving one equation for one of the variables
- Express the value of one variable in terms of the other (e.g., )
- Substitute this expression into the other equation and solve for the remaining variable
- Plug the value of the solved variable back into the original equation to find the value of the other variable
- involves manipulating the equations to eliminate one of the variables
- Multiply one or both equations by a to make the coefficients of one variable equal in magnitude but opposite in sign (e.g., and )
- Add the equations together to eliminate one variable and solve for the remaining variable
- Substitute the value of the solved variable into either original equation to find the value of the other variable
Solutions as line intersections
- The solution to a corresponds to the point where the two lines intersect on a graph
- If the lines intersect at a single point (e.g., ), the system has one
- The x-coordinate of the represents the value of the in the solution
- The y-coordinate of the point of intersection represents the value of the in the solution
- If the lines are parallel, they never intersect, indicating the system has no solution
- If the lines are coincident, they overlap completely, indicating the system has infinitely many solutions
Real-world applications of linear systems
- Identify the unknown quantities in the problem and assign them variables (e.g., let be the number of adults and be the number of children)
- Create two linear equations that represent the relationships between the unknown quantities based on the given information
- Solve the system of equations using graphing, substitution, or elimination method
- Interpret the solution in the context of the original problem and verify that it makes sense
- Examples of include:
- (e.g., determining the amounts of two solutions with different concentrations to mix to create a new solution with a specific concentration)
- (e.g., finding the prices of two items given the total cost of purchasing a combination of those items)
- (e.g., calculating the speeds of two objects traveling in opposite directions or in the same direction)
Modeling with Systems of Linear Equations
Understanding Linear Equations and Variables
- A is a mathematical expression that represents a straight line when graphed
- Slope represents the rate of change between two variables in a linear equation
- The (usually x) is the input value that can be freely chosen
- The (usually y) is determined by the value of the independent variable
Key Terms to Review (27)
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression or equation. It indicates how many times to scale the variable, influencing the overall value of the expression. Understanding coefficients is crucial in various mathematical contexts, as they help to establish relationships between variables and define equations in both algebraic and linear formats.
Coincident Lines: Coincident lines are two or more lines that lie exactly on top of each other in a two-dimensional plane, meaning they have all their points in common. This concept is significant because it indicates that the lines represent the same linear equation and thus have an infinite number of solutions, as every point on one line is also a point on the other. When graphically represented, coincident lines appear as a single line, which is essential in understanding systems of equations and their solutions.
Constant: A constant is a fixed value that does not change, regardless of the variables around it. In mathematical expressions, constants provide stability and serve as reference points, allowing for clear definitions and calculations. They are essential in forming equations, as they help differentiate between what is variable and what remains unchanged.
Coordinate plane: A coordinate plane is a two-dimensional surface formed by the intersection of a horizontal axis (x-axis) and a vertical axis (y-axis), allowing for the precise representation of points using ordered pairs. Each point on the coordinate plane is identified by its coordinates, which denote its position relative to the axes, facilitating the visualization and analysis of mathematical relationships and functions.
Cost problems: Cost problems involve analyzing and determining the total costs associated with various activities or production processes using mathematical models. These problems often require finding the optimal cost structures by using systems of linear equations, which represent relationships between different variables such as cost, quantity, and price.
Dependent variable: A dependent variable is a measurable factor that responds to changes in another variable, often referred to as the independent variable. It represents the outcome or effect that researchers are interested in observing, making it crucial for understanding relationships in various mathematical contexts. Its value depends on the input from the independent variable, highlighting its role in functions, systems of equations, and statistical analysis.
Distance-rate-time problems: Distance-rate-time problems are mathematical questions that involve calculating the relationship between distance traveled, speed (rate), and time taken. These problems typically utilize the formula $$d = rt$$, where 'd' represents distance, 'r' is the rate of speed, and 't' is time. Understanding this relationship allows for the solving of various real-world scenarios involving travel, motion, and logistics.
Elimination method: The elimination method is a technique used to solve systems of linear equations by removing one variable, allowing for the straightforward solution of the remaining variable. This method involves adding or subtracting equations to eliminate one of the variables, simplifying the process of finding the values of both variables in a system. It is particularly useful when dealing with larger systems where substitution may become cumbersome.
Graphing method: The graphing method is a visual technique used to solve systems of equations or inequalities by plotting their graphs on a coordinate plane. This method allows for the identification of solutions at the points where the graphs intersect, which represent the values that satisfy all equations or inequalities in the system. It is particularly effective for illustrating the relationships between variables and understanding how changes in one variable affect another.
Independent variable: An independent variable is a quantity that is manipulated or changed in an experiment or mathematical function to observe its effect on another quantity, called the dependent variable. In various contexts, it serves as the input for functions, the variable that is controlled in experiments, or the predictor in statistical models.
Infinitely many solutions: Infinitely many solutions refer to a situation in mathematical equations where there are countless valid solutions that satisfy the equation. This typically occurs in linear equations when the equations represent the same line or plane, leading to multiple points of intersection. Understanding this concept is crucial for solving and interpreting both single-variable and systems of linear equations.
Linear equation: A linear equation is a mathematical statement that describes a straight line when graphed on a coordinate plane, typically expressed in the form $$y = mx + b$$, where $$m$$ represents the slope and $$b$$ represents the y-intercept. This equation shows a direct relationship between two variables, indicating how one variable changes with respect to another. Linear equations can be manipulated and solved for unknowns, making them essential in various applications, from real-world problems to graphical representations.
Mixture problems: Mixture problems involve finding the quantities of different substances combined to form a new substance, often using systems of linear equations to determine the relationships between their proportions. These problems typically present real-world scenarios where two or more solutions or items with different properties are mixed, such as liquids or solids, and require the formulation of equations to solve for unknown quantities. The solutions often highlight the balance and constraints in the mixture.
No solution: No solution refers to a situation in which an equation or system of equations has no set of values that satisfies it. This can occur when the equations represent parallel lines in a graph, indicating that they never intersect and thus have no common solutions. Understanding this concept is vital for recognizing when certain problems cannot be solved within the given constraints, whether it's a single equation or a system involving multiple equations.
Ordered Pair: An ordered pair is a pair of elements where the order of the elements matters, typically represented as (x, y) in a coordinate system. The first element, x, is known as the x-coordinate, and the second element, y, is known as the y-coordinate. This concept is crucial for representing points in a two-dimensional space, which directly connects to solving equations and inequalities involving two variables.
Parallel lines: Parallel lines are lines in a plane that never meet or intersect, no matter how far they are extended. They maintain a constant distance apart and have the same slope when represented in a coordinate system, which is essential in understanding relationships between linear equations and geometric properties.
Point of intersection: A point of intersection is a specific point where two or more lines meet or cross each other on a graph. This point is crucial for solving systems of linear equations, as it represents the solution to the equations where both conditions are satisfied simultaneously. Identifying the point of intersection allows us to understand how the relationships between different variables are represented visually.
Real-world applications: Real-world applications refer to the practical use of mathematical concepts and methods to solve problems encountered in everyday life. These applications demonstrate how theoretical mathematics can be employed in various fields such as business, science, engineering, and social sciences, illustrating the relevance and utility of math in real situations.
Slope: Slope is a measure of the steepness or incline of a line, typically represented as the ratio of the vertical change to the horizontal change between two points on that line. It plays a crucial role in understanding relationships in equations and inequalities, helping to determine whether they increase or decrease, and is essential for graphing functions and analyzing systems of equations.
Solution set: A solution set is the collection of all values that satisfy a given equation or inequality. In mathematical contexts, this term is crucial as it helps identify all possible answers that make an equation true or meet the conditions of an inequality, creating a clearer understanding of relationships between variables.
Solutions of a system of equations: Solutions of a system of equations are the set of values that satisfy all equations in the system simultaneously. In a system of linear equations with two variables, these solutions correspond to the points where the graphs of the equations intersect.
Substitution method: The substitution method is a technique used to solve systems of linear equations by expressing one variable in terms of the other and substituting that expression into the second equation. This approach allows for finding the values of both variables in a straightforward manner, making it easier to analyze relationships within the equations. It is particularly useful when dealing with two-variable systems, simplifying the process of finding intersection points in graphs and determining feasible regions in optimization problems.
System of linear equations: A system of linear equations is a collection of two or more linear equations that share the same variables. The solutions to these equations are the points where the graphs of the equations intersect, which can represent various scenarios in real-world problems such as economics, physics, and engineering. Understanding how to solve these systems is crucial for analyzing relationships between quantities and making predictions based on those relationships.
Unique solution: A unique solution in the context of systems of linear equations refers to a specific outcome where there is exactly one pair of values that satisfies both equations simultaneously. This means the two lines represented by the equations intersect at a single point on the graph, which indicates that there is only one solution that makes both equations true. Understanding this concept is crucial as it distinguishes between various outcomes such as no solutions or infinitely many solutions, which are essential to grasp when analyzing linear systems.
X-variable: The x-variable is a symbol used to represent an unknown value in mathematical equations, particularly in the context of linear equations. In a system of linear equations, the x-variable is one of the two variables that determine the coordinates of points where lines intersect. Understanding the role of the x-variable is essential for solving systems of equations, graphing lines, and analyzing relationships between variables.
Y-intercept: The y-intercept is the point where a graph intersects the y-axis, representing the value of the dependent variable when the independent variable is zero. This key feature helps to understand linear relationships, curves, and data trends, providing crucial information for graphing and analyzing equations across various mathematical contexts.
Y-variable: The y-variable is one of the two variables in a system of linear equations, often representing the dependent variable whose value depends on the value of the x-variable. In the context of graphing linear equations, the y-variable is plotted on the vertical axis and is crucial for understanding the relationship between the two variables in the equation. It helps to determine how changes in the x-variable influence the y-variable and vice versa.