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Section 3.6 More Systems of Equations

Subsection 3.6.1 Overview

In the Homework 3.5.4, you determined how to solve a system of equations graphically. You also determined how to solve a system of equations algebraically when the equations were presented in the form, \(y = mx + b\text{.}\) In Section 3.6, you will play a game to hone your new-found skills then determine and analyze a system for which the equations are not in the form, \(y = mx + b\text{.}\)

Student Page 3.6.2 Linear Sovereignty

Play Linear Sovereignty until each player has had 3 turns. Use the gameboard in Figure  39 

Linear Sovereignty Rules.

On your turn:

  1. Draw a line that goes through two or more points on the gameboard. All lines must be functions. (What types of lines does this rule eliminate?)

  2. State a correct equation for the line you drew.

  3. Scoring is as follows. You must justify your score.

    • 1 point for each point on the gameboard that is on the line.

    • 1 point for algebraically determining the intersection point of your line with another line. The intersection point cannot be a point already on the gameboard.

    • 1 point for constructing a line parallel or perpendicular to an existing line.

  4. If an error is made in finding or justifying an equation or point of intersection, the player correcting the error steals the point(s).

  5. Play moves to the left. The game ends when each player has had three turns.

To Win.

The player with the most points at the end of the game wins.

Answer the following questions when you have finished playing Linear Sovereignty:


How did you determine that two lines were parallel?


Were you able to determine that two lines were perpendicular? How?


How can you determine the coordinates of a point of intersection?


Look at the graph again. How many pairs of parallel lines are possible?


How many pairs of perpendicular lines are possible?

Figure Linear Sovereignty Gameboard

Student Page 3.6.3 Measuring with Pretzles and Cheerios

Use pretzel sticks and cheerios to measure an 8.5-inch by 11-inch sheet of paper. The student page provides an interesting way to measure something if conventional tools are not available. As a bonus, you get to eat the pretzels and the cheerios when you're done measuring!  40 


Measure each side of an 8.5-inch by 11-inch sheet of paper using as many pretzel sticks as you can without exceeding the side length of the paper. Fill in the rest of the length using Cheerios. Record your values in Table

Table Measuring With Pretzels and Cheerios
Side of paper Side length (inches) Number of
pretzels, \(p\)
Number of
cheerios, \(c\)
Long side 11    
Short side 8.5    


For each side of the paper, write an equation to show the length in terms of the number of pretzels and the number of cheerios you needed to measure the side length.


Do the graphs intersect? If so, what is the point of intersection?


What do the coordinates of the intersection point represent in the context?


Measure the length of a pretzel stick and the diameter of a Cheerio. Compare your measurements to your solutions in Student Page Exercise Comment on your work based on these comparisons.

Homework 3.6.4 Homework


Solving a system of linear equations in two unknowns is an extension of the equation solving we have been doing all along when the equations are in the form \(y = mx + b\text{.}\)


Solve the system of equations below:

\begin{align*} y \amp = 4.2x - 1.6\\ y \amp = 1.7x + 3.8 \end{align*}

Indicate how the solution process is similar to what you have done in the past when solving equations.


Indicate what more you need to do to get a full solution to a system of equations.


In the cartoon, Foxtrot, Paige is frustrated by a system of equations problem she has to solve for homework. (See 41 , retrieved April 23, 2020.)

The system is:

\begin{align*} 2x + y \amp = 60\\ x + 2y \amp = 75 \end{align*}

Her older brother asked her, “If 2 shirts and a sweater costs $60 and a shirt and 2 sweaters costs $75, what does each item cost?” Paige solved the problem immediately but didn't realize she had done so.


What was Paige's answer?


Show Paige how to solve the system of equations in her homework. Use two different methods to solve the problem.

Linear Sovereignty is adapted from Linear Sovereignty by Trevor Rice and Charlene Beckmann in Adventures with Mathematics: Climbing from Grade 8 to Algebra 1, Michigan Council of Teachers of Mathematics, 2012, pp. 46—47.
Measuring with Pretzles and Cheerios is adapted from Pretzels & Cheerios by Tiffany Stob, in Adventures with Mathematics: Climbing from Algebra 1 to Geometry, Michigan Council of Teachers of Mathematics, 2010, pp. 19—20.