Elementary Algebra
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5.4 Solve Applications with Systems of Equations
Last Updated on June 24, 2024
Systems of equations are powerful tools for solving complex problems. They allow us to model relationships between multiple variables, representing real-world scenarios mathematically. This topic shows how to construct and apply these systems to various situations.
We'll learn to translate word problems into equations, tackle business and geometric challenges, and analyze uniform motion. These skills will help us solve intricate problems by breaking them down into manageable pieces and finding solutions systematically.
Constructing and Applying Systems of Equations
Systems of equations from word problems
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Top images from around the web for Systems of equations from word problems
Systems of Linear Equations: Two Variables – Algebra and Trigonometry OpenStax View original
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3.1 Systems of Equations – College Algebra for the Managerial Sciences View original
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- Identify unknown quantities and assign variables (x, y, z)
- Determine relationships between unknown quantities
- Look for key phrases indicating relationships ("the sum of", "the difference between", "twice as much as")
- Write equations representing relationships between variables
- Ensure number of equations matches number of unknown quantities
- Verify system of equations accurately represents given information in word problem
Real-world applications of systems
- Recognize situations modeled using systems of equations
- Business problems (profit, revenue, cost), mixture problems (combining solutions), investment problems (interest rates, principal)
- Assign variables to unknown quantities in problem
- Write equations based on given information and relationships between variables
- Solve system of equations using appropriate method (substitution, elimination, graphing)
- Interpret solution in context of original problem
- Verify solution makes sense and answers question posed in problem
- Consider real-world modeling constraints and limitations
Systems for geometric problem-solving
- Identify geometric properties and relationships given in problem
- Perimeter (sum of sides), area (length × width), angle relationships (supplementary, complementary)
- Assign variables to unknown quantities (lengths, angles)
- Write equations based on geometric properties and relationships
- Use formulas for perimeter, area, angle sums as needed
- Solve system of equations to find values of unknown quantities
- Interpret solution in context of geometric problem
- Verify solution satisfies given geometric properties and relationships
Uniform motion and systems of equations
- Understand relationship between distance, rate, time:
- Identify given information in problem (distances, rates, times)
- Assign variables to unknown quantities
- Write equations based on distance-rate-time relationship for each moving object or person
- Consider total distance traveled, sum of distances covered by each object or person
- Solve system of equations to find values of unknown quantities
- Interpret solution in context of uniform motion problem
- Verify solution makes sense and answers question posed in problem (time of intersection, distance traveled by each object or person)
Advanced system analysis techniques
- Variable isolation: Rearranging equations to express one variable in terms of others
- Graphical interpretation: Visualizing solutions as intersections of lines or curves
- System of inequalities: Extending systems to include constraints represented by inequalities
- Dependent variables: Identifying and handling relationships where one variable depends on others
Key Terms to Review (18)
Linear Equations: A linear equation is a mathematical equation in which the variables are raised to the first power and the variables are connected by addition, subtraction, multiplication, or division. Linear equations represent straight-line relationships between variables and are fundamental in solving various algebraic problems.
Solution Set: The solution set of an equation or inequality is the set of all values of the variable(s) that satisfy the given equation or inequality. It represents the complete set of solutions that make the statement true.
Variable Isolation: Variable isolation is the process of manipulating an equation to solve for a specific variable by isolating it on one side of the equation. This technique is essential in solving equations with fractions or decimals, as well as in solving applications involving systems of equations and mixture problems.
Mixture Problems: Mixture problems are a type of word problem that involve combining two or more substances or items with different properties, such as concentrations or costs, to create a new mixture with a specific desired property. These problems often require the use of systems of linear equations to find the unknown quantities in the mixture.
Ordered Pair: An ordered pair is a set of two numbers that represent a specific location on a coordinate plane. It consists of an x-coordinate and a y-coordinate, which together define a unique point in the rectangular coordinate system.
Slope-Intercept Form: Slope-intercept form is a way to represent the equation of a linear line in the form $y = mx + b$, where $m$ represents the slope of the line and $b$ represents the $y$-intercept, or the point where the line crosses the $y$-axis.
Substitution Method: The substitution method is a technique used to solve systems of linear equations by isolating one variable in one equation and substituting it into the other equation to find the values of the remaining variables. This method is particularly useful in solving systems of equations where the coefficients of the variables differ, allowing for the elimination of one variable through substitution.
Consistent System: A consistent system of linear equations is a set of equations that have at least one common solution, meaning the equations are compatible and can be satisfied simultaneously. This concept is central to solving systems of equations using various methods, including substitution, elimination, and applications.
Elimination Method: The elimination method is a technique used to solve systems of linear equations by eliminating one of the variables through addition or subtraction of the equations. This method allows for the determination of the values of the variables that satisfy all the equations in the system.
Simultaneous Equations: Simultaneous equations are a set of two or more equations that contain multiple variables and must be solved together to find the values of those variables. They are a fundamental concept in algebra and are used to solve a variety of problems, including those involving systems of equations, applications, and mixture applications.
System Matrix: The system matrix, also known as the coefficient matrix, is a mathematical representation of a system of linear equations. It is a matrix that contains the coefficients of the variables in the equations, and it is used to solve and analyze the system of equations.
Break-Even Analysis: Break-even analysis is a financial tool used to determine the point at which a company's total revenue equals its total costs, meaning it has neither a profit nor a loss. This analysis is particularly useful in the context of solving applications with systems of equations, as it helps businesses understand the relationship between their fixed costs, variable costs, and revenue to optimize their operations.
Cramer's Rule: Cramer's Rule is a method used to solve systems of linear equations by expressing the solution as a ratio of determinants. It provides a systematic way to find the values of the variables in a system of equations with the same number of equations as variables.
Graphical Interpretation: Graphical interpretation refers to the process of analyzing and understanding the meaning and implications of a visual representation, such as a graph or chart, in the context of a given problem or situation. It involves extracting relevant information, identifying trends, and drawing conclusions from the graphical data.
System of Inequalities: A system of inequalities is a set of two or more inequalities that must be satisfied simultaneously. It represents a collection of constraints or limitations that need to be considered together to find the feasible solutions that satisfy all the inequalities in the system.
Dependent Variables: Dependent variables are the variables in a system or equation that depend on or are influenced by the values of other variables, known as independent variables. They represent the outcomes or effects that are measured or observed in a study or experiment.
Real-World Modeling: Real-world modeling is the process of creating mathematical models that accurately represent and simulate real-life situations, problems, or phenomena. It involves translating real-world data, constraints, and relationships into a mathematical framework that can be analyzed and used to make informed decisions.
Intersection Point: The intersection point is the point where two or more lines, curves, or functions meet and share a common coordinate. It represents the solution to a system of equations, where the values of the variables satisfy all the equations simultaneously.