lesson
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If you buy tacos for 2eachanddrinksfor1 each, spending $10 total, there are several possible combinations—like 4 tacos and 2 drinks, or 3 tacos and 4 drinks.
Single linear equations have infinite combinations of (x,y) pairs that work. But what happens when we add a second rule—like knowing you bought 7 items in total?
What is a System of Equations?
A system of linear equations is a set of two or more linear equations that share the same variables.
A solution to a system is an ordered pair (x,y) that makes every equation in the system true at the same time.
📊An interactive-style visual card showing two equations: 'Condition 1: x + y = 7 (Total items)' and 'Condition 2: 2x + y = 10 (Total cost)'. Below it, three test pairs are displayed in boxes: Pair A (4, 3) -> Total = 7 (Pass), Cost = $11 (Fail); Pair B (2, 6) -> Total = 8 (Fail), Cost = $10 (Pass); Pair C (3, 4) -> Total = 7 (Pass), Cost = $10 (Pass). Pair C is highlighted with a glowing green checkmark and labeled 'The Solution'. Modern clean UI, light gray container, bold readable typography.
Pairs of numbers are easy to test with simple numbers, but what does this relationship look like on a coordinate grid?
The Graphical Meaning of a Solution
Every line on a coordinate plane represents all the infinite (x,y) points that satisfy that specific equation.
When you graph two lines together, their intersection point is the single coordinate that sits on both lines, making it the solution to the system.
📊A clean 2D Cartesian coordinate grid (x-axis from 0 to 8, y-axis from 0 to 8). Line 1 (blue) represents x + y = 7 passing through (0,7) and (7,0). Line 2 (purple) represents 2x + y = 10 passing through (0,10) and (5,0). The two lines cross cleanly at (3, 4). A prominent glowing green dot marks (3, 4) with callout text: 'Intersection (3, 4): Satisfies both lines simultaneously'.
How can you verify whether an ordered pair is truly a solution without taking the time to draw a graph?