The Defining Equation of a Line

Examples, solutions, worksheets, and videos to help Grade 8 students learn that two equations that graph as the same line are said to define the same line.

New York State Common Core Math Grade 8, Module 4, Lesson 23

Worksheets for Grade 8

Lesson 23 Student Outcomes

• Students know that two equations in the form of ax + by = c and ax’ + by’ = c’ graph as the same line when a’/a = b’/b = c’/c and at least one of a or b is nonzero.
  
• Students know that the graph of a linear equation ax + by = c, where a, b, and c are constants and at least one of a or b is nonzero, is the line defined by the equation ax + by = c.

Lesson 23 Student Summary
Two equations that graph as the same line are said to define the same line. Two equations that define the same line are the same equation, just in different forms. The equations may look different (different constants, different coefficients, or different forms).
When two equations are written in standard form, ax + by = c and ax’ + by’ = c’, they define the same line when a’/a = b’/b = c’/c is true.

Lesson 23 Opening Exercise
Exploratory Challenge/Exercises 1–3

1. Graph the equation 9x + 3y = 18 using intercepts. Then answer parts (a)–(f) that follow.
   a. Graph the equation y = -3x + 6 on the same coordinate plane.
   b. What do you notice about the graphs of 9x + 3y = 18 and y = -3x + 6? Why do you think this is so?
   c. Rewrite y = -3x + 6 in standard form.
   d. Identify the constants, a, b, c of the equation in standard form from part (c).
   e. Identify the constants of the equation 9x + 3y = 18. Note them as a’, b’, and c’.
   f. What do you notice about a’/a, b’/b and c’/c?
2. Graph the equation y = 1/2x + 3 using the y-intercept and the slope. Then answer parts (a)–(f) that follow.
   a. Graph the equation using 4x - 8y = -24 using intercepts on the same coordinate plane.
   b. What do you notice about the graphs of y = 1/2 x + 3 and 4x - 8y = -24? Why do you think this is so?
   c. Rewrite y = 1/2x + 3 in standard form.
   d. Identify the constants, a, b, c of the equation in standard form from part (c).
   e. Identify the constants of the equation 4x - 8y = -24. Note them as a’, b’, and c’.
   f. What do you notice about a’/a, b’/b and c’/c?
3. The equations y = 2/3 x - 4 and 6x - 9y = 36 graph as the same line.
   a. Rewrite y = 2/3 x - 4 in standard form.
   b. Identify the constants, a, b, c of the equation in standard form from part (a).
   c. Identify the constants of the equation 6x - 9y = 36. Note them as a’, b’, and c’.
   d. What do you notice about a’/a, b’/b and c’/c?
   e. You should have noticed that each fraction was equal to the same constant. Multiply that constant by the standard form of the equation from part (a). What do you notice?

Exercises 4–8
4. Write three equations that would graph as the same line as the equation 3x + 2y = 7.
5. Write three equations that would graph as the same line as the equation x - 9y = 3/4.
6. Write three equations that would graph as the same line as the equation -9x + 5y = -4.
7. Write at least two equations in the form ax + by = c that would graph as the line shown below.
8. Write at least two equations in the form ax + by = c that would graph as the line shown below.

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