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## Linear Programming

00:00

Write an inequality to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret data points as possible (i.e. a solution) or not possible (i.e. a non- solution) under the established constraints.

## Modeling One Variable Linear Equations

00:00

Solve linear equations in one variable.  Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation .

Write a function that describes a relationship between two quantities.  Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, simple rational, and exponential functions (integer inputs only).

## Solving Linear Equations

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Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.  Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

## Solving Linear Inequalities

00:00

Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.  Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

## Modeling with Linear Systems

00:00

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.  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.

## Solving Linear Systems by Graphing

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Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

## Modeling with Systems of Linear Inequalities

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Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 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.

## Modeling 1 Variable Linear Inequalities

00:00

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.  Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

## Solving Linear Systems by Substitution

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Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

## Solving Linear Systems by Multiplying

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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.