Unit 6: Systems of Linear Equations
Students explore what happens when you consider two linear equations simultaneously, and explore the many rich applications that can be modeled with systems of linear equations in two variables.
In Unit 6, eighth-grade students explore what happens when you consider two linear equations simultaneously. They graph two lines in the same coordinate plane and ask themselves what coordinate points satisfy both of the equations. They consider what it means when two lines never intersect or when they overlap completely. Students learn algebraic methods that can be used to solve systems when graphing is not efficient. Using the structure of the equations in a system, students will determine if systems have one, no, or infinite solutions without solving the system (MP.7). Students also explore the many rich applications that can be modeled with systems of linear equations in two variables (MP.4).
Students will use their knowledge from previous eighth-grade units, including work with single linear equations and functions from clusters 8.EE.B and 8.F.B. They will also need to draw on concepts from sixth grade, where they understood solving an equation as a process of answering which values make an equation true.
In high school, students will continue their work with systems, working with linear, absolute value, quadratic, and exponential functions. They will also graph linear inequalities and consider what the solution of a system of linear inequalities looks like in the coordinate plane.
Pacing: 15 instructional days (11 lessons, 3 flex days, 1 assessment day)
Fishtank Plus for Math
Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.
The following assessments accompany Unit 6.
Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.
Pre-Unit Student Self-Assessment
Have students complete the Mid-Unit Assessment after lesson 6.
Use the resources below to assess student mastery of the unit content and action plan for future units.
Post-Unit Assessment Answer Key
Use student data to drive your planning with an expanded suite of unit assessments to help gauge students’ facility with foundational skills and concepts, as well as their progress with unit content.
Suggestions for how to prepare to teach this unit
Prepare to teach this unit by immersing yourself in the standards, big ideas, and connections to prior and future content. Unit Launches include a series of short videos, targeted readings, and opportunities for action planning.
The central mathematical concepts that students will come to understand in this unit
Terms and notation that students learn or use in the unit
elimination (linear combinations)
substitution (to solve a system of equations)
system of linear equations
To see all the vocabulary for Unit 6, view our 8th Grade Vocabulary Glossary.
Topic A: Analyze & Solve Systems of Equations Graphically
Define a system of linear equations and its solution.
Solve systems of linear equations by graphing.
Classify systems of linear equations as having a unique solution, no solutions, or infinite solutions.
Solve real-world and mathematical problems by graphing systems of linear equations.
Create a free account to access thousands of lesson plans.
Already have an account? Sign In
Topic B: Analyze & Solve Systems of Equations Algebraically
Solve systems of linear equations using substitution when one equation is already solved for a variable.
Solve systems of linear equations using substitution by first solving an equation for a variable.
Solve real-world and mathematical problems using linear systems and substitution.
Solve systems of linear equations using elimination (linear combinations) when there is already a zero pair.
Solve systems of linear equations using elimination (linear combinations) by first creating a zero pair.
Solve real-world and mathematical problems using systems and any method of solution.
Model and solve real-world problems using systems of equations.
The content standards covered in this unit
— Analyze and solve pairs of simultaneous linear equations.
— Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
— Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
— Solve real-world and mathematical problems leading to two linear equations in two variables.
For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
Standards covered in previous units or grades that are important background for the current unit
— Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
— Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
— Solve linear equations in one variable.
— Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Standards in future grades or units that connect to the content in this unit
— Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
— Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
— Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
— Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
For example, find the points of intersection between the line y = -3x and the circle x² + y² = 3.
— Represent a system of linear equations as a single matrix equation in a vector variable.
— Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
— Make sense of problems and persevere in solving them.
— Reason abstractly and quantitatively.
— Construct viable arguments and critique the reasoning of others.
— Model with mathematics.
— Use appropriate tools strategically.
— Attend to precision.
— Look for and make use of structure.
— Look for and express regularity in repeated reasoning.
Pythagorean Theorem and Volume