Lesson 1: 6-1 Graphing Systems of Equations

If two lines have the same slope, but different y-intercepts, how many times do they intersect?

If two lines have different slopes, how many times do they intersect?

If two lines have the same slope and same intercept, how many times do they intersect?

system of equations
consistent
independent
dependent
inconsistent

A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

SAT questions related to systems: 4-3-16,1-3-18,1-3-9,2-3-2,8-3-18,6-4-11,7-3-3,1-3-11,7-4-11,1-4-19,8-3-10,2-4-29,2-3-20,3-3-9

A system of equations can be solved by graphing the equations on the same coordinate plane. The graphs of the equations can intersect at one point (exactly one solution), be parallel (no solution), or be the same line (infinite number of solutions).

Bicycling Naresh rode 20 miles last week and plans to ride 35 miles per week. Diego rode 50 miles last week and plans to ride 25 miles per week. Predict the week in which Naresh and Diego will have ridden the same number of miles.

For Exercise 48, suggest that students try several prices at both discounts, such as $50, $100, and $150, and then decide whether Francisca or Alan is correct.

Interpersonal Learners: Have students work in pairs or groups to check the solutions for Exercises 16-24 and 27-38. Suggest that they use the Study Tip (p. 340) on comparing m and b when both equations are in the form of y = mx + b. Have students write equations in slope-intercept form, if necessary.

Extension: Write a system of three equations in two variables on the board. Have the students determine if the system has one solution, no solution, or infinitely many solutions. If it has one solution, name it. For example,

x + y = 2

x - y = 0

y = -2

has no solution because the three lines do not intersect at one point.

Practice: Exercises 1 -9

Exercises 53-59

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