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3 Responses
Nice graph work! To pin the intersection without guessing, set the y-values equal because at the crossing they match: 2x + 1 = −x + 4. That gives 3x = 3, so x = 1, and then y = 2(1) + 1 = 3. So the exact intersection is (1, 3). You can also see this directly from your plotted points by following the slopes: from (0,1) with slope 2, go right 1 and up 2 to (1,3); from (0,4) with slope −1, go right 1 and down 1 to (1,3). Unless I’m misreading your grid’s scale, that lands cleanly at (1, 3), not around (1.2, 3.6).
A quick mental formula when lines are in y = mx + b form is x = (b2 − b1)/(m1 − m2). Here, m1 = 2, b1 = 1, m2 = −1, b2 = 4, so x = (4 − 1)/(2 − (−1)) = 3/3 = 1, then plug back to get y = 3. If you want a short refresher on the substitution/elimination approach (the no-graph way), this Khan Academy lesson is helpful: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/v/solving-systems-of-equations-by-substitution.
Set the y’s equal: 2x+1 = -x+4 ⇒ 3x=3 ⇒ x=1, then y=3, so the lines intersect at (1,3), which should land exactly on a grid point. Want a quick elimination method you can do mentally for any two lines?
A quick, no-guess way is to set the two lines equal because at the crossing they share the same y-value-so I think you just solve 2x + 1 = -x + 4 and that pins down the x right from the grid numbers. If I add x to both sides I get 3x + 1 = 4, and then (I might be mixing a sign here?) subtract 1 to get 3x = 2, so x = 2/3, and then y = 2(2/3) + 1 = 7/3, about 2.33. That feels a bit off compared to your eyeball 3.6, but the idea is you don’t need to guess: equate the equations, solve for x, then plug back to get y. Worked example: for y = 2x + 1 and y = −x + 4, I’m getting x = 2/3 and y ≈ 2.33 by that little two-step. On a grid, you can also “walk” the slopes from the intercepts-start at (0,1), go up 2 and right 1 repeatedly, and from (0,4) go down 1 and right 1-where those stair-steps meet is the same point I just found (I think?). Anyway, the takeaway is: equal the y’s, solve for x, then back-substitute for y-no more squinting at the crossing like it’s a smudged treasure map.