Lesson 1: General Form: Ax+By=c

Which variable represents slope in the slope-intercept form?

Which variable represents the y-intercept in the slope-intercept form?

Which variable represents the x-intercept in the slope intercept form?

Input
output
slope
vertical intercept
horizontal intercept
slope intercept form
general form

A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.

Lesson Description This lesson focuses on the structure and manipulation of the equation ax + by = c. Unlike slope-intercept form, this version places both variables on one side, which requires different techniques for analysis. Key skills include: Identifying the General Form: Recognizing equations where x and y are on the same side and identifying the coefficients a, b, and the constant c. Converting to Slope-Intercept Form: Using algebraic literal equation skills to "solve for y." This is a critical bridge for students who want to find the slope or use a graphing calculator. Graphing via Intercepts: Learning the "Cover-Up Method" to find the x-intercept (let y=0) and the y-intercept (let x=0). This provides two easy points to plot without needing to rearrange the equation. Working with Integers: Managing equations where a, b, and c are integers, which often simplifies the process of finding intercepts.
Purpose
The purpose of Section 11.1 is to develop algebraic flexibility. In the real world, many relationships naturally occur in General Form—for example, a budget constraint like 5x + 10y = 100 (where x is the number of 5 items and y is the number of $10 items). By mastering this form, students learn that there is more than one way to represent a line and that the "best" form depends on what information you are trying to find. This section also reinforces the solving techniques required for systems of equations in later courses.
Depth of Knowledge (DOK) Level
DOK Level 2
Level 2 (Skill/Concept): Students must perform multi-step transformations. Converting ax + by = c to y = mx + b requires multiple operations (subtraction followed by division). Similarly, finding two intercepts and plotting them involves a coordinated sequence of substituting zeros, solving, and graphing. It requires students to choose the most efficient method (intercepts vs. solving for y) based on the numbers provided.

If some students have difficulty with word problems because they cannot picture what the problem is trying to communicate, then sometimes it is easier for those students to graph or draw a picture of the given information before writing the equation. For Guided Practice 5, you may wish to have students do part b first by using the starting point and the rate of change to determine other points on the graph. Then have students write the equation that describes the line formed.

Extension Write 3x + 2y = 8 and -3x + 2y = 8 on the board. Remind students that these equations are in the standard form for the equation of a line. Ask students to tell how the equations are alike and how they are different. Then, ask students to tell how the graphs of these two equations are alike and how they are different.

practice problems page 269-272

Review on page 272

Unit assessment