Solve real-world and mathematical problems that lead to pairs of linear equations

Learning goal: Solve real-world and mathematical problems leading to pairs of linear equations in two variables, including deciding whether the line through one pair of points intersects the line through another pair. The solutions here are limited to those that can be found or addressed by graphing, with substitution used to confirm what the graph shows.

From a situation to a pair of equations. Many problems describe two quantities that each change at a steady rate from a starting value. Each such quantity is a linear equation in two variables, and the question "when are they equal?" is a pair of simultaneous linear equations. The strategy is: define the variables with units, write each equation as y = mx + b by identifying the starting value b and the rate m, graph both lines on the same axes, read the intersection, and check the point in both equations and in the story.

A fully worked model: two gym plans. Suppose, hypothetically, a gym offers Plan A for a 24-dollar sign-up fee plus 4 dollars per visit, and Plan B for 7 dollars per visit with no fee. Let v be the number of visits and C the total cost in dollars. Plan A: C = 24 + 4v, intercept 24 and slope 4. Plan B: C = 7v, intercept 0 and slope 7. On a graph with visits on the horizontal axis and cost on the vertical axis, Plan A starts higher but rises more slowly; Plan B starts at the origin and rises faster. The lines cross at (8, 56). Check: 24 + 4(8) = 24 + 32 = 56 and 7(8) = 56. Both plans cost 56 dollars for 8 visits. The graph also answers the practical question: for fewer than 8 visits, Plan B's line is lower, so it is cheaper; for more than 8, Plan A's line is lower. At 5 visits, A costs 44 and B costs 35; at 12 visits, A costs 72 and B costs 84. The variable v counts whole visits, so the lines are a model and only whole-number points describe real bills.

Lines through two pairs of points. A purely mathematical version asks: given coordinates for two pairs of points, does the line through the first pair intersect the line through the second pair? Compute each slope (a pair of points with the same x-coordinate gives a vertical line, which has no slope; a vertical line meets every non-vertical line exactly once, and two different vertical lines never meet). If the slopes differ, the lines intersect at exactly one point, and graphing finds or estimates it. If the slopes are equal, either the lines are parallel and distinct, with no intersection, or they are the same line, and checking whether a point of one lies on the other tells which. Worked: Line 1 passes through (1, 2) and (3, 8): slope (8 − 2) ÷ (3 − 1) = 3, and from (1, 2) the intercept is 2 − 3 = −1, so y = 3x − 1. Line 2 passes through (0, 5) and (4, 7): slope 2 ÷ 4 = 1/2 and intercept 5, so y = (1/2)x + 5. The slopes 3 and 1/2 differ, so the lines intersect. Graphing shows the crossing near x = 2.4, y = 6.2. Confirm: 3(2.4) − 1 = 6.2 and (1/2)(2.4) + 5 = 1.2 + 5 = 6.2, so (2.4, 6.2) is exact. The algebra agrees: 3x − 1 = (1/2)x + 5 gives (5/2)x = 6 and x = 12/5.

When two-point lines do not intersect. Line 3 passes through (0, 1) and (2, 5): slope 2, equation y = 2x + 1. Line 4 passes through (1, 6) and (3, 10): slope 2, and from (1, 6) the intercept is 6 − 2 = 4, so y = 2x + 4. Equal slopes and different intercepts mean parallel lines that never meet: no intersection. Had Line 4 passed through (3, 7) and (5, 11) instead, its slope would still be 2 but the intercept would be 7 − 6 = 1, making it the same line as Line 3, with infinitely many shared points.

Reading the graph in context. The intersection answers "when are the two quantities equal?" The regions on either side answer "which is larger before and after?" For quantities that grow over time, the steeper line overtakes the other after the intersection, as long as it started below it. For costs, the lower line is cheaper. State the answer with units and in a sentence about the situation, not only as an ordered pair. When the intersection is not on a grid point, report an estimate from the graph, then substitute to see how close the estimate is; if an exact answer is needed, set the two expressions for y equal and solve the resulting one-variable equation, which is exactly the skill from earlier in this unit.

Misconceptions to avoid. Confusing the rate with the starting value is the most common modeling error: in C = 24 + 4v, the 4 is the slope and the 24 is the intercept, and swapping them produces a wrong graph. Another is reading the intersection's x-coordinate as the answer to every question; sometimes the question asks for the cost, which is the y-coordinate, or for a range of visits, which is a region of the graph. A third is assuming two lines that both rise must meet; two non-vertical lines meet at a single point only if their slopes differ, and with equal slopes they are either parallel and distinct or the same line; a vertical line meets every non-vertical line once, and two vertical lines are either distinct and parallel or the same line.

Worked two pairs of points: Line 1 through (0, 0) and (2, 6): slope 3, equation y = 3x. Line 2 through (0, 4) and (4, 8): slope 1, equation y = x + 4. The slopes differ, so the lines intersect. Graphing shows the crossing at (2, 6). Check: 3(2) = 6 and 2 + 4 = 6. Notice that (2, 6) is one of the given points of Line 1, which is allowed; the given points are simply two points that fix each line.

Worked sum and difference: Two numbers have sum 10 and difference 4. With x the larger and y the smaller: x + y = 10 and x − y = 4. Rewritten for graphing, y = −x + 10 and y = x − 4. The first has intercept 10 and slope −1; the second has intercept −4 and slope 1. They cross at (7, 3). Check: 7 + 3 = 10 and 7 − 3 = 4. The numbers are 7 and 3.

Worked two tanks: Tank 1 holds 50 liters and drains at 4 liters per minute: V = 50 − 4t. Tank 2 holds 10 liters and fills at 6 liters per minute: V = 10 + 6t. Graph both with minutes on the horizontal axis and liters on the vertical axis. The falling line and the rising line cross at (4, 34). Check: 50 − 4(4) = 34 and 10 + 6(4) = 34. After 4 minutes both tanks hold 34 liters; before that Tank 1 holds more, after that Tank 2 holds more. The model applies only while 0 ≤ t ≤ 12.5, because Tank 1 is empty at 50 ÷ 4 = 12.5 minutes.

Worked parallel two-point lines: Line through (0, 1) and (2, 5): slope 2, y = 2x + 1. Line through (1, 6) and (3, 10): slope 2, y = 2x + 4. Same slope, different intercepts, so the graphs are parallel and the pair has no solution. The second line is always 3 units above the first.

Representation task: Graph the gym plans C = 24 + 4v and C = 7v on the same axes for 0 to 12 visits. Mark the intersection and verify it in both equations. Shade the visits for which Plan B is cheaper and, in a different color, the visits for which Plan A is cheaper. Write a two-sentence recommendation that names the number of visits at which the plans cost the same, 56 dollars at 8 visits, and explains why only whole numbers of visits describe real bills.

Independent performance: Write one hypothetical real-world situation that leads to a pair of linear equations (two savings accounts, two phone plans, two vehicles with different head starts and speeds) and one purely mathematical problem giving two pairs of points. For each, define the variables with units, write both equations, graph them, decide whether the lines intersect, and find or estimate the intersection. Confirm the intersection by substitution in both equations, and for the real-world case state what the intersection and the regions on either side mean in the story. Then alter one number in the real-world situation so that the lines become parallel, and explain what that means in context. Success criteria: slopes and intercepts are identified from the situation correctly; the graph and the substitution agree; any estimate is labeled and its accuracy is reported; the contextual interpretation names the quantity, the units and which option is larger or cheaper on each side of the intersection.