Students repeatedly graph multiple linear equations on the same coordinate plane and compare their lines (Activity 1, Activity 2, Activity 3, Activity 4). Several activities and answer keys show multiple lines plotted together with intersection points highlighted (for example, Problem Set 1 shows three lines meeting at (0,2); Activity 3 includes a vertical line x=3 graphed with other lines). Students use slope-intercept form, plot y-intercepts, and extend lines so that where lines cross is visible on the graphs.

Students write and graph linear cost equations for car, train, and plane (for example y = 0.15x + 31.50 and y = 0.20x + 20.75) and plot these lines on a shared graph. The activity explicitly asks when driving becomes cheaper than the train and the answer key has students set the two equations equal (0.15x + 31.50 = 0.20x + 20.75) to find x = 215, then interprets that value as the point where costs are equal. Students also graph distance vs. time lines and compare rates, practicing both graphical and algebraic representations of linear relationships.

Students work with graphs that display two lines on the same axes (e.g., Gym Memberships, Bike Race, Lemonade Stand, Growth Rate of Plants) and answer questions such as "When do both gym memberships cost the same?" and "When are both riders at the same location?". Several activity pages prompt students to read where two lines meet and use that point in answering questions about equal values. The answer keys for those activities indicate specific times or points where the two situations have the same value, showing students identify intersection points visually.

Students graph pairs of linear equations and are explicitly instructed to identify the point of intersection as the solution (examples include y = 2x + 1 and y = -x + 4 with solution (1, 3), and multiple activity pages that ask for the coordinate pair when lines intersect). The lesson has students verify intersection points by substitution: they substitute (1, 3) into both y = 2x + 1 and y = -x + 4 and show both equations are true, and similarly substitute (1, -2) into 2x - y = 4 and 4x - 2y = 8 to show every point on the line satisfies both equations. Several activities require students to label systems as one solution, no solution, or infinite solutions and to predict intersections from slopes by converting equations to slope-intercept form.

The lesson explicitly states that "The solution to a system of equations is the point where both equations are true at the same time" and that "When a system of equations is graphed, any point where the lines intersect is a solution." Students are asked to graph pairs of lines (for example, 3x - 2y = 7 and x + y = 4 with the intersection labeled (3, 1)) and then solve the same system algebraically to compare results. Multiple activities require students to check their candidate solution by substituting the x and y values back into both original equations to verify both equations are true simultaneously.

The lesson explicitly tells students that an intersection point is a solution: Step 3 of the example explains that (1, 4) lies on both lines and shows plugging x = 1 into each equation yields y = 4. Multiple student activity pages require students to graph two lines from two given points, estimate or identify their intersection, and write the intersection as an ordered pair (e.g., several graphs labeled with one/no/infinite solutions and blank grids for plotting). The Intersection Challenge and accompanying answer key have students find equations from two points and then solve algebraically (substitution or elimination) to produce the same ordered-pair intersection, reinforcing that the intersection satisfies both equations.

Students set up and solve pairs of linear equations from word problems using substitution and elimination (e.g., Sophie and Jake, Candy Shop, multiple activity problems). Students record solutions as coordinate pairs (for example, (30, 35) and (4, 3.85)) and explicitly solve systems by equating two y-expressions (e.g., y = 20x + 50 and y = 30x leading to 30x = 20x + 50). Students are also prompted to create graphs to "see a picture of the solution" and are told that the break-even point is the solution where both lines give the same cost.

The Skills section explicitly states: "Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously." Multiple Student Activity Pages ask students to graph pairs of lines and record the solution (for example, Problem 12: graph y = -x + 4 and y = x - 2 and write Solution: (3, 1); Problems 24 and 25 require finding equations from two points and solving or classifying intersections). The answer key and images label intersection points as solutions and show parallel lines labeled "no solution," and some solutions are found by setting the two equations equal algebraically, tying the intersection to satisfying both equations.

Students write linear equations for two options and graph both lines on the same axes in multiple parts of the project (Housing, Transportation, Entertainment, Meal Plans, Phone Plans). Students are instructed to set the two equations equal and solve algebraically to find the break-even point (e.g., 1200m = 1500 + 1050m; 225 + 0.60x = 1.25x) and then mark the intersection on the graph. In the Phone Plans activity students observe that two algebraically equivalent equations produce the same line and are told this yields infinite solutions, reinforcing the connection between equation solutions and graph intersections.

Students are asked to graph pairs of linear equations on the same coordinate plane (e.g., Plan A: y = 20x and Plan B: y = 15x + 30; Job A: y = 18x and Job B: y = 12x + 20) and to compare which line has the greater rate of change and which option costs more after a given time. Several activities include plotting two lines together and images show lines that intersect, and students are prompted to read values from those graphs. Students write equations for two competing linear situations and use the graphs to compare outcomes at specific x-values.

The lesson's Skills list explicitly states that students should "understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously." Activity 3 has students solve systems by graphing (e.g., y = x + 1 and y = -x + 5; y = 2x - 4 and y = -x + 2) where students locate the intersection points as solutions. Activity 2 has students determine whether equations have one, no, or infinitely many solutions and graph lines in slope-intercept form, linking algebraic cases to graphical behavior.

Students are asked to "Solve the system by graphing: y = x + 2, y = -x + 6" (Problem 32) and the provided graph shows the two lines intersecting at (2, 4) with the answer key listing (2, 4). Students also solve systems using substitution (Problem 33) and elimination (Problem 34), requiring them to find values of x and y that satisfy both equations simultaneously.