11.1 Systems of Linear Equations: Two Variables
3 min read•june 24, 2024
Linear systems are a powerful tool for solving real-world problems. They involve multiple equations with shared variables, representing lines on a graph. The solution is where these lines intersect, giving us valuable insights into complex situations.
We can solve linear systems through graphing, substitution, or elimination methods. Each approach has its strengths, allowing us to tackle different types of problems efficiently. Understanding these techniques opens doors to solving a wide range of practical challenges.
Solving Systems of Linear Equations
Graphing solutions of linear systems
Top images from around the web for Graphing solutions of linear systems
Solve Systems of Linear Equations with Two Variables – Intermediate Algebra View original
Is this image relevant?
Solving Systems of Equations by Graphing | College Algebra View original
Is this image relevant?
Solve a system of linear equations | College Algebra View original
Is this image relevant?
Solve Systems of Linear Equations with Two Variables – Intermediate Algebra View original
Is this image relevant?
Solving Systems of Equations by Graphing | College Algebra View original
Is this image relevant?
1 of 3
Top images from around the web for Graphing solutions of linear systems
Solve Systems of Linear Equations with Two Variables – Intermediate Algebra View original
Is this image relevant?
Solving Systems of Equations by Graphing | College Algebra View original
Is this image relevant?
Solve a system of linear equations | College Algebra View original
Is this image relevant?
Solve Systems of Linear Equations with Two Variables – Intermediate Algebra View original
Is this image relevant?
Solving Systems of Equations by Graphing | College Algebra View original
Is this image relevant?
1 of 3
- contains two or more linear equations with same variables
- Each equation represents a line on coordinate plane
- Solution is point(s) where lines intersect
- and of intersection point satisfy both equations simultaneously
- To graph system of linear equations:
- Graph each equation as line on same coordinate plane
- Identify point(s) of intersection (-intercept, -intercept)
- Number of solutions can be:
- One solution: Lines intersect at single point ()
- No solution: Lines are parallel and do not intersect ()
- Infinitely many solutions: Lines are coincident or overlapping ()
- The of each line plays a crucial role in determining the type of solution
Substitution and elimination methods
- solves for one variable in terms of other variable
- Solve one equation for one variable ( or )
- Substitute expression for variable into other equation
- Solve resulting equation for remaining variable
- Substitute value of solved variable back into expression to find value of other variable
- eliminates one variable by adding equations together
- Multiply one or both equations by constant to make coefficients of one variable equal in magnitude but opposite in sign
- Add equations together to eliminate one variable
- Solve resulting equation for remaining variable
- Substitute value of solved variable into one of original equations to find value of other variable
Consistency of linear systems
- has one solution (intersecting lines) or infinitely many solutions (coincident lines)
- has no solution (parallel lines)
- To determine type of system:
- If lines intersect at single point, system is consistent with one solution
- If lines are parallel ( and ), system is inconsistent
- If lines are coincident ( and ), system is consistent with infinitely many solutions
Solutions for dependent systems
- has infinitely many solutions
- Equations in system are equivalent, representing same line
- Solution can be expressed as ordered pair where and are
- Example: If solution is , any value of will satisfy both equations
Real-world applications of linear systems
- Identify unknown quantities and assign variables
- Create system of linear equations based on given information and relationships between variables
- Solve system using substitution, elimination, or graphing method
- Interpret solution in context of real-world problem
- Verify solution makes sense and satisfies given conditions (positive values, integers)
- If no solution or infinitely many solutions, explain what it means in context of problem
- No solution: Problem has no valid answer (insufficient information)
- Infinitely many solutions: Problem has multiple valid answers (dependent relationship between variables)
Matrix Methods for Solving Systems
- Systems of linear equations can be represented as
- An can be used to represent the system compactly
- is a method for solving systems using matrix operations
- Involves row operations to transform the augmented matrix into row echelon form
Key Terms to Review (25)
Augmented Matrix: An augmented matrix is a special type of matrix that is used to represent a system of linear equations. It is formed by combining the coefficient matrix of the system with the column of constants on the right-hand side of the equations.
Consistent system: A consistent system is a set of equations that has at least one solution. In a graph, the lines representing the equations intersect at one or more points.
Consistent System: A consistent system is a set of linear equations that has at least one solution. In other words, a consistent system is one where the equations can be satisfied simultaneously, meaning there exists a set of values for the variables that make all the equations true.
Dependent system: A dependent system is a system of linear equations in which all equations represent the same line, resulting in infinitely many solutions. This occurs when the equations are scalar multiples of one another.
Dependent System: A dependent system, in the context of linear equations, refers to a system where the equations are linearly dependent, meaning that one equation can be expressed as a linear combination of the other equations. This implies that the system has an infinite number of solutions or no solution at all.
Elimination Method: The elimination method, also known as the method of elimination, is a technique used to solve systems of linear equations by systematically eliminating variables to find the unique solution. This method is applicable in the context of various topics, including parametric equations, systems of linear equations in two and three variables, and systems of nonlinear equations and inequalities.
Gaussian elimination: Gaussian elimination is a method for solving systems of linear equations. It transforms the system's augmented matrix into row-echelon form using row operations.
Gaussian Elimination: Gaussian elimination is a method for solving systems of linear equations by transforming the system into an equivalent one that is easier to solve. It involves a series of row operations on the augmented matrix of the system to obtain an upper triangular matrix, which can then be used to find the solution to the system.
Inconsistent system: An inconsistent system is a set of equations that has no solution. This typically occurs when the equations represent parallel lines that never intersect.
Inconsistent System: An inconsistent system is a system of linear equations that has no solution, meaning there is no set of values for the variables that satisfies all the equations simultaneously. This term is particularly relevant in the context of solving systems of linear equations in two or more variables, as well as the techniques of Gaussian elimination and Cramer's rule.
Linear System: A linear system is a collection of linear equations that describe the relationship between multiple variables. These equations can be solved simultaneously to find the values of the variables that satisfy all the equations in the system.
Parametric Variables: Parametric variables are independent variables that can be manipulated or adjusted to explore how a system or function behaves. They are often used in mathematical models and equations to represent the input values that can be varied to observe the corresponding changes in the output or dependent variables.
Point of Intersection: The point of intersection is the location where two or more lines, curves, or surfaces intersect. It represents the common solution or point where the equations describing these geometric entities meet.
Point-slope formula: The point-slope formula is a method used to find the equation of a line given a point on the line and its slope. It is expressed as $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.
Simultaneous Equations: Simultaneous equations are a set of two or more equations that share common variables and must be solved together to find the values of those variables. They are a fundamental concept in algebra and are essential for understanding systems of linear and nonlinear equations.
Slope: Slope is a measure of the steepness or incline of a line or a surface. It represents the rate of change between two variables, typically the change in the vertical direction (y-coordinate) with respect to the change in the horizontal direction (x-coordinate).
Substitution method: The substitution method is a technique for solving systems of equations by substituting one equation into another. This transforms the system into a single-variable equation that can be solved more easily.
Substitution Method: The substitution method is a technique used to solve systems of linear equations, systems of nonlinear equations, and other types of equations by substituting one variable in terms of another. This method involves isolating a variable in one equation and then substituting that expression into the other equation(s) to solve for the remaining variable(s).
System of Linear Equations: A system of linear equations is a collection of two or more linear equations that share the same set of variables. These equations must be solved simultaneously to find the values of the variables that satisfy all the equations in the system.
X-coordinate: The x-coordinate is the first value in an ordered pair $(x, y)$ representing a point's horizontal position on the Cartesian plane. It indicates how far left or right the point is from the origin (0, 0).
X-Coordinate: The x-coordinate is the horizontal position of a point on a coordinate plane. It represents the distance from the origin (0,0) to the point along the horizontal x-axis. The x-coordinate is a crucial component in understanding and working with various mathematical concepts, including coordinate systems, graphs, unit circles, and systems of linear equations.
X-intercept: The x-intercept is the point where a graph crosses the x-axis, where the y-coordinate is zero. It represents the solution(s) to an equation when $y = 0$.
X-Intercept: The x-intercept of a graph is the point where the graph of a function or equation intersects the x-axis, indicating the value of x when the function's output or the equation's value is zero. The x-intercept is a crucial concept in understanding the behavior and properties of various mathematical functions and equations.
Y-coordinate: The y-coordinate is the vertical position of a point on a coordinate plane, measured as the distance from the x-axis. It represents the up-down position of a point and is used to describe the location of objects or data points within a two-dimensional coordinate system.
Y-intercept: The y-intercept is the point at which a line or curve intersects the y-axis, representing the value of the dependent variable (y) when the independent variable (x) is zero. It is a crucial concept in understanding the behavior and properties of various mathematical functions and their graphical representations.