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2 Responses
I totally get the temptation to circle (0,0) because it looks like the “home base” of the graph-but the real intersection is the spot where both lines give the same y for the same x, like two friends agreeing on the same café at the same time. Instead of setting the slopes equal (that would only tell you they’re parallel), set the equations equal: 2x + 1 = −x + 4, so 3x = 3, x = 1, and then y = 2(1) + 1 = 3-so they meet at (1, 3). On the graph, you can plot the y-intercepts first (1 for the first line, 4 for the second), then use the slopes to step: up 2 over 1 for y = 2x + 1, and down 1 over 1 for y = −x + 4; you’ll see them cross at (1,3). Since one slope is 2 and the other is −1, they’re negative reciprocals, so the lines are perpendicular and make a neat “X” shape-plus, a quick reality check shows (0,0) can’t be right because plugging x = 0 into y = 2x + 1 gives y = 1, not 0. If you want a visual walkthrough, this Khan Academy explainer on solving systems by graphing is great: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-equations/solving-systems-graphically/v/solving-systems-by-graphing-1. Want to try another pair and see if you can eyeball the intersection just from intercepts and slopes?
I’ve totally had that “ooh, the origin looks tidy, let’s pick that” moment! On the graph, the intersection is the point where the two drawn lines actually cross-where they share the same x and the same y at the same time. You don’t want to set the slopes equal (that’s how you check if lines are parallel; equal slopes usually means they never meet). A quick plotting trick: y = 2x + 1 starts at (0,1) and with slope 2 you go up 2 and right 1; y = −x + 4 starts at (0,4) and with slope −1 you go down 1 and right 1, and you’ll see them meet. Algebraically, at the intersection their y-values are equal, so set 2x + 1 = −x + 4, which gives 3x = 3, so x = 1 and then y = 2(1) + 1 = 3. So the real meeting point is (1,3). I sometimes try to “average the intercepts” to guess a crossing (which, uh, isn’t a real rule and can mislead), but setting the equations equal is the reliable way.