# Lesson 4 Some of One, None of the Other Solidify Understanding

## Learning Focus

Graph linear equations in standard form.

Why is it useful to use equivalent forms of linear equations, and how do I convert a linear equation from one form to the other?

## Open Up the Math: Launch, Explore, Discuss

Carlos and Clarita are comparing strategies for writing equations of the boundary lines for the* Pet Sitters* constraints. They are discussing their work on the **space** constraint.

**Space**: Cat pens will requireof space, while dog runs require . Carlos and Clarita have up to available in the storage shed for pens and runs, while still leaving enough room to move around the cages.

Carlos’s Method: “I made a table. If I don’t have any cats, then I have room for

Clarita’s Method: “I let

### 1.

Since both equations represent the same information, they must be equivalent to each other.

#### a.

Show the steps you could use to turn Clarita’s equation into Carlos’s equation. Explain why you can do each step.

#### b.

Show the steps you could use to turn Carlos’s equation into Clarita’s. Explain why you can do each step.

### 2.

Use both Carlos’s and Clarita’s methods to write the equation of the boundary line for the start-up costs constraint.

#### a.

Carlos’s method:

#### b.

Clarita’s method:

Start-up costs: Carlos and Clarita plan to invest much of the

### 3.

Show the steps you could use to turn Clarita’s start-up costs equation into Carlos’s equation. Explain why you can do each step.

### 4.

Show the steps you could use to turn Carlos’s start-up costs equation into Clarita’s. Explain why you can do each step.

Pause and Reflect

In addition to writing an equation of the boundary lines, Carlos and Clarita need to graph their lines on a coordinate grid.

Carlos’s equation is written in slope-intercept form. Clarita’s equation is written in standard form. Both forms are ways of writing linear equations.

Both Carlos and Clarita know they only need to plot two points in order to graph a line.

### 5.

Carlos’s strategy: How might Carlos use his slope-intercept form,

### 6.

Clarita’s strategy: How might Clarita use her standard form,

### 7.

Write equations for the following two constraints:

Space:

Start-up costs:

Find where the two lines intersect algebraically. Record enough steps so that someone else can follow your strategy.

### 8.

What does this point mean in the context of cats and dogs?

## Ready for More?

If you only know the

## Takeaways

Since two points determine a line, one strategy for graphing a linear equation is:

This strategy works particularly well for linear equations written in .

Another strategy for graphing a linear equation is:

This strategy works particularly well for linear equations written in .

Contexts that are best represented by linear equations in standard form contain information about: .

Contexts that are best represented by linear equations in slope-intercept form contain information about: .

## Adding Notation, Vocabulary, and Conventions

The standard form for a linear equation is, by convention

To change slope-intercept form to standard form:

To change standard form to slope-intercept form:

## Vocabulary

- linear function
- slope-intercept form of a line
- standard form of line
**Bold**terms are new in this lesson.

## 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.

Graph each of the linear inequalities on the coordinate grid. Check a point to make sure you correctly shaded the half plane containing the solutions.

### 1.

### 2.

### 3.

What methods do you have for finding the solution to a system of equations?