# Unit 5Systems of Equations and Inequalities

## Lesson 1

### Learning Focus

Solve systems of linear equations that represent a context.

### Lesson Summary

In this lesson, we reviewed strategies for solving systems of linear equations, including using tables, graphs, and algebra. We examined two different algebraic methods, both of which involve substitution of expressions for variables. These methods work with systems of linear equations that are linear functions, as well as systems of linear equations that are linear combinations.

## Lesson 2

### Learning Focus

Represent linear constraints.

### Lesson Summary

In this lesson, we encountered a situation in which decisions were limited by certain constraints. We tested combinations of quantities to see if they fit one, both, or neither constraint. Although we didn’t resolve the situation today, we recognized the goal of this type of work: to make decisions within the parameters of the constraints that will maximize (or minimize) a quantity, such as profit.

## Lesson 3

### Learning Focus

Graph the solution set for linear inequalities in two variables.

### Lesson Summary

In this lesson, we learned how to find and represent all of the points in the solution set for a linear inequality in two variables.

## Lesson 4

### Learning Focus

Graph linear equations in standard form.

### Lesson Summary

In this lesson, we learned the conventions for writing the standard form of a linear equation and strategies for turning slope-intercept form into standard form and standard form into slope-intercept form. We also learned a new method for graphing linear equations in standard form by finding the intercepts.

## Lesson 5

### Learning Focus

Represent constraints symbolically and graphically.

### Lesson Summary

In this lesson, we drew upon everything we have learned so far about modeling contexts with linear inequalities as constraints. We learned that in a system of constraints, quantities need to be measured in the same units, and that individual constraints are equivalent and produce the same solution set even if the quantities are measured in different units.

## Lesson 6

### Learning Focus

Find the solution set for a system of linear inequalities.

### Lesson Summary

In this lesson, we learned how to find the solution to a system of inequalities by finding the points in the coordinate plane that satisfy all of the inequalities simultaneously. We also used the solution to the system of inequalities for the pet sitter constraints to find the feasible region of viable options for Carlos and Clarita to consider when starting their business.

## Lesson 7

### Learning Focus

Solve systems of linear inequalities.

### Lesson Summary

In this lesson, we learned how to differentiate between linear inequalities in two variables written as strict inequalities, inequalities with or signs, and other inequalities written with or . We also practiced finding the solution set for systems of linear inequalities.

## Lesson 8

### Learning Focus

Develop intuitive strategies for solving systems of linear equations in standard form.

### Lesson Summary

In this lesson, we developed a strategy for finding the price of individual items from knowing the total cost of purchasing different combinations of those items. We realized we could invent purchases that weren’t actually made by doubling or tripling a known purchase or adding two purchases together. Finding two purchases in which the number of one of the items purchased was the same gave us insights into how to find the price of each individual item.

## Lesson 9

### Learning Focus

Solve systems of linear equations by eliminating one of the variables.

### Lesson Summary

In this lesson, we learned how to solve systems of equations by eliminating one of the variables. To do so, we had to think of equations as objects that can be added, subtracted, or multiplied by a scale factor. Each operation creates an equivalent system of equations, and if we are strategic, we can get a system of equations for which the solutions to the system are apparent.

## Lesson 10

### Learning Focus

Solve systems of equations.

Identify systems that have no solutions or an infinite number of solutions.

### Lesson Summary

In this lesson, we learned that systems of linear equations can have infinitely many solutions, no solutions, or one solution, and we learned how to identify which type of solution to expect by examining the coefficients of the linear equations in the system when they are written in standard form. We also extended the process of solving systems to a system of linear equations with three unknowns.

## Lesson 11

### Learning Focus

Solve systems using row reduction of matrices.

### Lesson Summary

In this lesson, we learned another method to solve systems of linear equations besides graphing, substitution, and elimination. To do so, we represented the system of equations with augmented matrices, and solved the system by using the process of row reduction of matrices.

## Lesson 12

### Learning Focus

Apply a standard procedure for row-reduction of matrices.

### Lesson Summary

In this lesson, we examined a standard procedure for the row reduction of matrices. The procedure eliminated all of the guesswork of solving a system of equations with matrices, but still allowed room for sense-making and strategic thinking. We also learned notation for recording our steps in the solution process, rather than having to describe how we got one matrix from a previous one in words.