# Linear Equations, Inequalities and Systems

## • Explain each step in solving a simple equation in one variable (A-REI.A.1). • Create and solve linear equations and inequalities in one variable, including equations with coefficients represented by letters (A-CED.A.1$^\star$, A-REI.B.3). • Model constraints and relationships between quantities by equations and inequalities, and by systems of equations and inequalities, and interpret solutions (A-CED.A.2$^\star$, A-CED.A.3$^\star$). • Solve systems of linear equations approximately by graphing and exactly by algebraic methods (A-REI.C.6). • Understand the principles behind the method of elimination (A-REI.C.5). • Graph the solution set to a linear inequality in two variables as a half-plane (A-REI.D.12). • Graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes (A-REI.D.12).

The story before this unit:

Students begin learning about ratios and rates in Grade 6. In Grade 7, they represent proportional relationships by equations of the form $y = kx$, understanding $k$ as the constant of proportionality or unit rate. In Grade 8, they recognize such equations as special kinds of linear equations $y = mx + b$ where $m$ is the constant of proportionality and $b$ is 0. They understand $m$ as the slope of the line obtained from graphing the equation and $b$ as the $y$-intercept of the line which is the value of $y$ when $x = 0.$

By this point in their mathematical trajectory, then, students should be fairly comfortable with linear equations in two variables. They should be able to create and graph such equations to represent real-world situations that can be modeled by linear equations. Just as importantly, students should be able to describe the fundamental characteristic of linear functions, namely that they have a constant rate of change: the change in the output variable is proportional to the corresponding change in the input variable.

In grade 8, students analyzed and solved pairs of simultaneous linear equations (8.EE.C.8). Students should:
• know that the solutions to a system of two linear equations in two variables correspond to points of intersections of their graphs, because points of intersection satisfy both equations simultaneously (8.EE.C.8.a);
• know how to solve a system of two linear equations in two variables algebraically (8.EE.C.8.b);
• be able to estimate solutions by graphing the equations (8.EE.C.8.b);
• know how to solve real-world and mathematical problems leading to a system of two linear equations in two variables (8.EE.C.8.c).

The part of the story happening in this unit:

In this unit, students build on what they know from middle school about linear equations and inequalities and systems of linear equations and expand their understanding to include systems of linear inequalities. They work with more complex modeling problems and become fluent in general methods of solution. They make more sophisticated use of graphical methods of representing and solving equations, inequalities, and systems, and they interpret points or regions in the plane in terms of the context.

The story after this unit:

Students will apply what they learn in this unit to the study of bivariate statistics. They will revisit the notion of a function and use the techniques learned here to study linear functions. They will come to view linear functions as one of many function families that display predictable characteristics. The technique of substitution, learned to find values that simultaneously satisfy two linear equations, is useful in more general situations, e.g., to solve a system consisting of a linear and a quadratic equation (A-REI.7) and in differential and integral calculus. For students who study calculus, linear functions will be an essential basis for their work with derivatives and differentiation.

## Sections

A1.2.0 Pre-unit diagnostic assessment

#### Summary

Assess students’ ability to
•  solve linear equations in one variable with rational number coefficients (8.EE.C.7.b);
•  solve word problems leading to linear inequalities of the form of the form $px + q \geq r$ with rational coefficients (7.EE.B.4.b);
•  graph linear equations of the form $y = mx + b$ (8.F.A.3);
•  solve a real-world problem that leads to a simple case of a system of two linear equations in two variables, where the same variable occurs with coefficient 1 in both equations (8.EE.C.8).

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A1.2.1 Overview of linear equations and inequalities in two variables

#### Summary

Create and graph the solutions of linear equations and inequalities in two variables, and discuss their meaning in a real-world context (A-CED.A.2$^\star$, A-CED.A.3$^\star$, A-REI.B.3, A-REI.D.12).

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A1.2.2 Reason about linear equations and inequalities in one variable

#### Summary

•  Explain each step in solving a simple equation in one variable (A-REI.A.1).
•  Create and solve linear equations in one variable, including equations with coefficients represented by letters (A-REI.B.3).
•  Create and solve linear inequalities in one variable (A-REI.B.3).

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A1.2.3 Model with systems of linear equations

#### Summary

•  Model constraints and relationships between quantities with systems of linear equations (A-CED.A.2$^\star$, A-CED.A.3$^\star$).
•  Solve systems of linear equations approximately by graphing and exactly by algebraic methods (A-REI.C.6).

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A1.2.4 Mid-unit assessment

#### Summary

Assess students’ ability to
•  solve equations and inequalities in one variable (A-REI.B.3);
•  explain each step in solving a simple equation (A-REI.A.1);
•  solve systems of equations graphically and algebraically (A-REI.C.6);
•  set up and solve systems of equations that model a context (A-CED.A.3$^\star$);
•  interpret a solution to a system of equations in terms of the context (A-CED.A.3$^\star$).

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A1.2.5 Solve general systems of linear equations in two variables

#### Summary

•  Solve systems of linear equations exactly by algebraic methods (A-REI.C.6).
•  Understand the principles behind the method of elimination (A-REI.C.5).

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A1.2.6 Model with inequalities in two variables

#### Summary

•  Identify constraints from a context, choose relevant variables and model the context with an inequality or system of inequalities (A-CED.A.3$^\star$).
•  Identify coordinates pairs or points in the plane as solutions or non-solutions and interpret them in terms of the context (A-CED.A.3$^\star$).
•  Graph solution sets to a linear inequality or system of inequalities (A-REI.D.12).

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A1.2.7 Summative assessment

#### Summary

Assess students’ ability to
•  create and solve linear equations and inequalities in one variable (A-REI.B.3);
•  solve systems of linear equations exactly by algebraic methods (A-REI.C.6);
•  model relationships between quantities and compare different relationships (A-CED.A.2$^\star$, A-REI.C.6);
•  graph the solution set to a linear inequality in two variables as a half-plane, and to a system of linear inequalities in two variables as the intersection of the corresponding half-planes (A-REI.D.12).

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