Section 3.5 : Standard Form

Learning Targets: A.CED.2, F.IF.4, F.IF.7a and F.IF.9

Important Terms and Definitions

Standard Form: 𝐴𝑥 + 𝐵𝑦 = 𝐶, where A, B, C are integers

x – intercept : the x – coordinate of the point where a graph crosses the x – axis

y – intercept : the y – coordinate of the point where a graph crosses the y – axis

Converting from Standard Form to Slope-Intercept Form

Just solve for y!

Example: Rewrite 2𝑥 + 6𝑦 = 18 in slope-intercept form.

2𝑥 + 6𝑦 = 18

6𝑦 = −2𝑥 + 18

𝑦 = − 26

𝑥 + 186

𝑦 = − 13

𝑥 + 3

(ex 1) Rewrite 4𝑥 + 12𝑦 = −18 in slope-intercept form.

(ex 2) Rewrite 7𝑥 − 2𝑦 = 4 in slope-intercept form.

Finding the x and y – intercepts

Example: Find the x and y – intercepts of the equation 5𝑥 + 2𝑦 = −10.

x – intercept : Let 𝑦 = 0 y – intercept : Let 𝑥 = 0

5𝑥 + 2(0) = −10 5(0) + 2𝑦 = −10

5𝑥 = −10 2𝑦 = −10

𝑥 = −2 𝑦 = −5

(−2, 0) (0, −5)

(ex 3) Find the x and y – intercepts of the equation 3𝑥 + 5𝑦 = 15

(ex 4) Find the x and y – intercepts of the equation 2𝑥 + 5𝑦 = 6

(ex 5) Find the x and y – intercepts of the equation −5𝑥 + 3𝑦 = −7.5

Graphing From Standard Form

Option 1: Find the x and y – intercepts and graph them

Option 2: Solve for y so that the equation is in slope-intercept form. Then graph using the same steps used in Section 3-3

(ex 6) Graph −2𝑥 + 𝑦 = −6 using x and y – intercepts

(ex 7) Graph −2𝑥 + 3𝑦 = 15 by putting the equation in slope-intercept form.

Explore: Do all lines have both x and y intercepts?

Graphing Vertical and Horizontal Lines

𝑥 = # → vertical line 𝑦 = # → horizontal line

(ex 8) Graph 𝑦 = 3 (ex 9) Graph 𝑥 = −4 (ex 10) Graph 𝑦 = −1

Converting to Standard Form

Get the x and y to the same side, then simplify so that the equation includes only integers.

(ex 11) 𝑦 = 7𝑥 + 2

(ex 12) 𝑦 = 23

𝑥 + 6

(ex 13) 𝑦 = 56

𝑥 + 13

Using the Standard Form as a Model

(ex 14) Buckle is having a sale! Each t-shirt will be $12 and long sleeved t-shirts will be $15 each. You plan to spend $120 on t-shirts and long sleeved t-shirts. Write and graph an equation that represents this situation. What are three combinations of t-shirts and long sleeved t-shirts can buy for $120?

Homework – Section 3.5 : Standard Form

Rewrite each equation in slope-intercept form and use it to graph.

1. 2𝑥 + 7𝑦 = 14 3. 6𝑥 − 12𝑦 = 4 2. −𝑥 + 4𝑦 = −8 4. −4𝑥 − 24𝑦 = −36

Find the x and y – intercepts of the following equations and use them to graph.

5. −8𝑥 + 10𝑦 = 40 7. −6𝑥 + 3𝑦 = −9 6. 7𝑥 − 2𝑦 = 14 8. 4𝑥 + 12𝑦 = −18

Graph each line.

9. 𝑦 = −8 11. 𝑦 = 43

10. 𝑥 = 3.5 12. 𝑥 = −1

Convert the following to standard form, using only integers.

13. 𝑦 = 5𝑥 − 12 15. 𝑦 = − 25

𝑥 + 110

14. 𝑦 = − 34

𝑥 − 4 16. 𝑦 = 47

𝑥 + 43

17. Mark runs at an average pace of 8 mi/h and walks at an average rate of 3 mi/h. Define variables for the amount of time he spends running and the amount of time he spends walking. Then write an equation in standard form that relates the amount of time he spends running and walking if he travels a distance of 15 miles.

18. You are preparing a post-workout snack mix to take with you to track meets. You plan to include peanuts and granola, and want each mix to have 28 grams of protein. Peanuts have 7 grams of protein per ounce and granola has 3 grams of protein per ounce. Write an equation to represent the protein content of your snack mix. Graph the equation, and use that graph to determine the number of ounces of granola you should include if you plan to use 1 ounce of peanuts.