Modeling with Systems of Linear Equations
Build and solve real-world systems of linear equations to model scenarios with two unknown quantities using substitution or elimination in Grade 9 Algebra.
Key Concepts
Property Given a situation that represents a system of linear equations, write the system of equations and identify the solution. 1. Identify the input and output of each linear model. 2. Identify the slope and $y$ intercept of each linear model. 3. Find the solution by setting the two linear functions equal to one another and solving for $x$, or find the point of intersection on a graph. This point is where the models have equal value.
Examples Company A rents a car for 30 dollars a day plus 25 cents per mile. Company B charges 50 dollars a day plus 15 cents per mile. To find when they cost the same, set $30 + 0.25d = 50 + 0.15d$. Solving gives $0.10d = 20$, so $d=200$ miles.
Phone Plan A is 40 dollars per month. Plan B is 20 dollars per month plus 5 dollars per GB of data. They cost the same when $40 = 20 + 5g$, which means $g=4$ GB. For more than 4 GB, Plan A is cheaper.
Common Questions
What is Modeling with Systems of Linear Equations?
Modeling with Systems of Linear Equations is a key concept in Grade 7 math. It involves applying specific rules and properties to simplify expressions, solve equations, or analyze mathematical relationships. Understanding this topic builds foundational skills needed for higher-level algebra and beyond.
How is Modeling with Systems of Linear Equations used in real-world applications?
Modeling with Systems of Linear Equations appears in practical contexts such as financial calculations, engineering problems, and data analysis. Mastering this skill helps students model and solve problems they will encounter in science, technology, and everyday decision-making situations.
What are common mistakes when working with Modeling with Systems of Linear Equations?
Common errors include forgetting to apply rules to all terms, sign errors when working with negatives, and skipping verification steps. Always double-check by substituting answers back into the original problem and reviewing each algebraic step carefully.