➕ Algebra 1 · Systems of Equations
Solving Systems by Graphing: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Meet at the corner
Jo and Kai are meeting downtown. On a city map with an x-axis running east and a y-axis running north, Jo texts, “I’ll be on Main Street,” which is the vertical line x = 3. Kai texts, “I’ll be on 5th Avenue,” which is the horizontal line y = 5. Only one spot is on both streets: the corner (3, 5). If Kai stands at (8, 5), that is on 5th Avenue but not on Main Street, so the two have not met.
How it connects: Each text is an equation, and a spot that satisfies both is the solution of the system. The spot (8, 5) satisfies only one equation, which is why you check a possible solution in both. The limit of the picture: streets at right angles are a special case. Slanted lines, like y = 2x + 1, also cross in exactly one point whenever their slopes are different. Two different streets that run the same direction never meet, which is the no-solution case.
Two receipts, two unknown prices
At a food truck, let x be the price of one taco and y the price of one drink, both in dollars. Receipt 1 says 2 tacos and 1 drink cost 8 dollars: 2x + y = 8. Receipt 2 says 1 taco and 2 drinks cost 7 dollars: x + 2y = 7. Receipt 1 alone fits many price pairs, such as (1, 6) and (2, 4), but neither fits Receipt 2, since 1 + 2(6) = 13 and 2 + 2(4) = 10. Only one pair fits both receipts: tacos cost 3 dollars and drinks cost 2 dollars, the point (3, 2).
How it connects: Each receipt is a line (all the price pairs that fit it), and the real prices are where the two lines cross. The limit of the picture: prices cannot be negative, so only part of each line is used. And if Receipt 2 were just Receipt 1 doubled (4 tacos and 2 drinks for 16 dollars), it would be the same line and add no new information, so infinitely many price pairs would fit both.
The two-lock vault
In a made-up video game, a vault has two locks and two dials, x and y. Lock A opens whenever the dials satisfy y = x + 4. Lock B opens whenever they satisfy y = −x + 12. The vault door opens only when both locks are open. Try the dials (1, 5): Lock A opens because 5 = 1 + 4, but Lock B stays shut because −1 + 12 = 11, not 5. Try (4, 8): 8 = 4 + 4 and 8 = −4 + 12, so both locks open and the door unlocks.
How it connects: Each lock is one equation. A dial setting on only one line, like (1, 5), opens only one lock, so a possible solution must be tested in both equations. The limit of the picture: it is a made-up game. A system whose lines are parallel is a vault that can never open (no solution). A system that gives the same line twice is a vault where every setting that opens one lock also opens the other (infinitely many solutions).
Lane lines in a pool
Look at the black lane lines painted on the bottom of a swimming pool. They run in the same direction, side by side, and stay the same distance apart. Follow them as far as you like: they never meet.
How it connects: Same direction means same slope. Staying apart means different y-intercepts. Parallel lines never meet, so a system whose lines are parallel has no solution. The limit of the picture: a pool’s lane lines are short segments, while a line in math goes on forever. And two lines with the same slope and the same y-intercept are one line, like painting a line on top of itself. They share every point, so that system has infinitely many solutions.
Look before you leap
The proverb advises you to think about what you are doing before you jump into it.
How it connects: Before you announce a crossing point, look: substitute it into both equations. A graph can be slightly off, and a point that fits only one equation is not a solution of the system. The limit of the proverb: it is general advice about caution. Here it means one specific action, testing the pair in both original equations.
Parallel lines at the cafe
Two parallel lines sat down at a cafe. “We have so much in common,” said one. “Same slope, same direction.” “True,” said the other, “but we will never meet.”
How it connects: Parallel lines have the same slope and different y-intercepts, so they never meet: the system has no solution. The limit of the joke: having the same slope is not enough to guarantee no solution. If the y-intercepts match too, the two lines are one line, and they share every point.
Memory help
GRID
Get each equation ready to graph (usually y = mx + b). Ruler-draw both lines on one grid (plot (0, b), then use the slope). Identify the crossing point and read (x, y). Double-check it in both equations.
Cross, parallel, same
“Cross once, one solution found. Parallel lines, no common ground. Same line twice, solutions abound.” This is for lines you can write as y = mx + b: crossing means different slopes, parallel means the same slope with different b, and the same line means the same slope and the same b.
Slope first, then b
Compare the slopes first. Different slopes: the lines cross once, so there is one solution. Same slopes: now look at b. Different b means side by side (no solution). The same b means one line drawn twice (infinitely many solutions).
A solution has two numbers
The solution of a system is a pair (x, y). Saying “x = 1” alone is half an answer. And (3, 1) is a different point from (1, 3), because x always comes first.
One line is not enough
A point on only one line makes only one equation true. Test every possible solution in both equations before you write it down.
Circle, label, check
Draw the two lines in two colors and write each equation next to its line. Circle the crossing point and write its coordinates beside the circle. Then write two short lines underneath: “Equation 1: true” and “Equation 2: true.”