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 require $6 {ft}^{2}$ of space, while dog runs require $24 {ft}^{2}$ . Carlos and Clarita have up to $360 {ft}^{2}$ 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 $15$ dogs. If I use some of the space for $4$ cats, then I can have $14$ dogs. With $8$ cats, I have room for $13$ dogs. For each additional dog run that I don’t buy, I can buy $4$ more cat pens. From my table I know the $y$ -intercept of my line is $15$ and the slope is $- \frac{1}{4}$ , so my equation is $y = - \frac{1}{4} x + 15$ .”

Clarita’s Method: “I let $x$ represent the number of cats, and $y$ the number of dogs. Since cat pens require $6 {ft}^{2}$ , $6 x$ represents the space used by cats. Since dog runs require $24 {ft}^{2}$ , $24 y$ represents the amount of space used by dogs. So, my equation is $6 x + 24 y = 360$ .”

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 $$ 1, 280$ they earned from their last business venture to purchase cat pens and dog runs. It will cost $$ 32$ for each cat pen and $$ 80$ for each dog run.

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, $y = - \frac{1}{4} x + 15$ , to plot two points on his line?

6.

Clarita’s strategy: How might Clarita use her standard form, $6 x + 24 y = 360$ , to plot two points on her line? (Clarita is really clever, so she looks for the two easiest points she can find.)

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 $x$ - and $y$ -intercepts of a linear function, how can you find both the standard form and the slope-intercept form of the equation of the line?

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

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.

Retrieval

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.

$y > - \frac{1}{4} x + 5$

2.

$y \leq 2 x - 8$

3.

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