I’m trying to solve simultaneous equations by graph, and my lines keep choosing extremely unromantic meeting spots: somewhere between the squares, like they’re shy about committing to an actual lattice point. My graph paper now looks like a waffle of indecision.
Example 1: y = 2x + 3 and y = -x + 5. These are already in slope-intercept form (nice!), so I plotted (0,3) and (1,5) for the first, and (0,5) and (1,4) for the second. The lines cross somewhere that looks like x is a bit more than 0.6 and y is a little over 4. But depending on how I tilt my head, the intersection scoots a smidge. I suspect I’m not choosing a good scale or I’m eyeballing badly.
Example 2: 2x + 3y = 12 and y – x = 1. I rearranged to y = -2/3 x + 4 and y = x + 1. I plotted both using points from quick tables. My attempt got an intersection around x ≈ 1.7-ish, y ≈ 2.7-ish, but when I try to be more precise with a ruler, the crossing drifts. I think my scale (1 square = 1 unit on both axes) makes the gentle slope too flat and it amplifies tiny drawing errors.
Extra confusion: when I tried x = 3 and y = 2x – 1, I wasn’t sure if I’m allowed to just draw a vertical line at x = 3 without rearranging anything. I did that and got an intersection by sight, but I felt like I was breaking a rule.
What I’m stuck on:
– How do I pick a sensible scale so the intersection is readable, especially when it lands at fractional coordinates?
– Is there a reliable way to estimate the intersection from a graph well enough to write it as a fraction, or is that expecting too much from a hand-drawn graph?
– How many points should I plot per line to avoid wobble? I usually do two, but maybe that’s too optimistic with steep/shallow slopes.
– Any tips for catching parallel or coincident lines before I waste time drawing them, and for dealing with one vertical/one slanted line cleanly?
If someone could point out where my approach is going squiggly (scale? plotting points? reading the crossing?), I’d love to make my lines meet like polite, punctual lines instead of elusive cartesian cryptids.
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3 Responses
When lines play hard-to-get between squares, I let algebra nail the exact meetup and then pick a friendlier scale (so slopes aren’t pancake-flat), plotting at least three points per line and checking slopes first to catch parallels; verticals like x=3 are totally fine. Example: y=2x+3 and y=-x+5 meet where 2x+3=-x+5 -> x=2/3, y=13/3 (use 1 square = 1/3 to see it), and x=3 with y=2x-1 meets at (3,5); similarly, y=-2/3 x+4 and y=x+1 cross at x=9/5, y=14/5.
Short version: graphing is great for a visual check, but it rarely gives exact fractional intersections unless the point lands on your grid. For an exact answer, solve the system algebraically, then use the graph to verify and to see geometry. Example 1: 2x + 3 = −x + 5 gives 3x = 2, so x = 2/3 and y = 13/3 ≈ 4.33 – exactly what your sketch suggested. If you want the graph to “show” 2/3 or 13/3, pick a scale that marks thirds (e.g., 1 square = 1/3). When plotting, two points define a line, but for accuracy use three well-spaced points and prefer “clean” ones: intercepts or x-values that make the slope’s denominator divide nicely. For y = −2/3 x + 4, take x = 0, 3, 6 to get (0,4), (3,2), (6,0); for y = x + 1, use (−1,0), (2,3), (4,5). Draw the full-length line with a straightedge; short segments magnify small errors.
Choose a scale that zooms into where the lines meet: center the region, and don’t feel bound to “1 unit per square.” If you expect fifths, use fifths; if thirds, use thirds. If one slope is very gentle, compress the other axis or enlarge that direction so the crossing isn’t almost parallel to a grid line. Expect a hand-drawn reading error around a tenth or two of a unit; writing exact fractions from a graph alone is usually asking too much unless your grid matches the denominators. Quick diagnostics: in y = mx + b form, equal slopes mean parallel lines; equal slopes and equal intercepts mean the same line. In Ax + By = C form, lines are parallel if A/B matches, and identical if A:B:C are proportional. Vertical lines are fine: x = 3 is a legitimate line; with y = 2x − 1 the intersection is (3, 5) by direct substitution.
Simple worked example (your Example 2). Solve 2x + 3y = 12 and y − x = 1 by substitution: y = x + 1, so 2x + 3(x + 1) = 12 ⇒ 5x + 3 = 12 ⇒ x = 9/5 and y = 14/5. To see that cleanly on paper, plot y = −(2/3)x + 4 at x = 0, 3, 6 and y = x + 1 at x = −1, 2, 4, and use a scale with fifths marked; the lines will meet at (1.8, 2.8).
Short answer: your plotting is fine; your scale and point choices are doing you dirty. For exact answers, don’t expect a hand sketch to magically spit out fractions unless you set the grid to match them. Example 1 solves to x = 2/3, y = 13/3. If you want to see that cleanly on a graph, pick a scale that marks thirds (say, 1 big square = 3 small ticks, each tick = 1/3). Then both 2/3 and 13/3 sit right on grid lines and the crossing stops “scooting.” Example 2 solves to x = 9/5, y = 14/5, so mark fifths and you’ll nail it. Practical rule: if the algebra coughs up denominators like 3 or 5, subdivide your axes to that denominator. If the lines are too shallow or nearly parallel, stretch one axis (different x- and y-scales are totally allowed) so the crossing isn’t crammed in a tiny angle. And yes, x = 3 is a perfectly legal vertical line-draw it straight up at 3 and read the other line’s y there; algebraically you’re just plugging x = 3 into y = 2x − 1 to get y = 5.
Plotting tips that cut wobble: two points define a line, but use two far-apart, “clean” points. Intercepts are great: for 2x + 3y = 12, (6,0) and (0,4) are miles apart-your ruler errors shrink. For slope-intercept form, start at the intercept and “step the slope” using rise/run with integers (for 2/3, go right 3, up 2) to land on another lattice point. I usually mark three points and make sure they’re collinear before I commit the line. To spot time-wasters early: equal slopes with different intercepts means parallel (no solution); same slope and same intercept means the same line (infinitely many). Quick check from Ax + By = C: if A1:B1 = A2:B2 but C ratios don’t match, they’re parallel. By the way, graphs are for seeing what’s going on; exact fractions come from the algebra. I learned this the hard way in ninth grade-kept “refining” a sketch that said x ≈ 1.7 until my teacher sighed, handed me graph paper with fifths, and, boom, the dot snapped to 1.8. Ever since, I pick the scale to fit the fractions, not my ruler’s mood.