Algebra 1
Solving by graphing
Start with what a single equation already gives you. Write y = x + 1, and you don't get one answer. You get a whole line of them: (0, 1), (1, 2), (2, 3), and on forever. Write y = −x + 5, and that's a second line, its own endless list of points. Now ask a fair question: is there a single point that lives on both lines at the same time?
That question is what a system asks. Two relationships, both true at once. The answer, when there is one, is the point where the two lines cross 7.1.f1. It's the one (x, y) that satisfies both equations together.
The way to see it is to graph both lines on one plane and look for where they meet. One caution before you draw, though: don't trust your eye to read a crossing point off the picture. Compute the points instead. Pick a few x-values, run each through both equations, and watch for the row where the two outputs match. That row is the crossing.
Take y = x + 1 and y = −x + 5. Here's a short table for each:
$$\begin{array}{c|c|c} x & y = x+1 & y = -x+5 \\ \hline 0 & 1 & 5 \\ 2 & 3 & 3 \\ 4 & 5 & 1 \end{array}$$
Look at the row x = 2: both equations give y = 3. So both lines pass through (2, 3), and that shared point is the solution. The graph would show the two lines crossing right there. You didn't need to eyeball it, though, because the table told you exactly.
New words
- 7.1.d1 System of equations: two (or more) equations considered together, asking for values that satisfy all of them at once.
- 7.1.d2 Solution of a system: an ordered pair (x, y) that makes every equation in the system true. A system may have one such pair, none, or infinitely many (see 7.4). So don't assume there's exactly one until you've solved it.
Notice that a solution here is an ordered pair, not a single number. This is the one thing to hold onto for the whole unit: you're not done when you know x, you're done when you have both x and y. Finding x = 2 and walking away is finishing half the problem.
Worked example
Read each of these slowly, line by line, and ask why each step follows from the one before.
7.1.w1 Example 1, the canonical one. Solve y = x + 1 and y = −x + 5 by graphing. From the table above, both lines pass through (2, 3), so that's the only point on both. Now confirm it by putting (2, 3) into both equations, not just one, because the answer has to satisfy both at once. First equation: is 3 = 2 + 1? Yes. Second equation: is 3 = −2 + 5? Yes. Both are true, so the solution is (2, 3).
If a check like that ever doesn't come out true, you haven't failed. Your check just did its job and caught something before it counted. That's the whole reason we check. Go back to your first step and re-run the arithmetic slowly. A mismatch is almost always one sign or one small slip, not the whole method falling apart.
7.1.w2 Example 2, a horizontal line. Solve y = 2x and y = 6. The equation y = 6 is a flat line sitting at height 6. Every point on it has y = 6, no matter the x. The line y = 2x climbs steadily, and you want to know when it reaches that same height of 6. It does when 2x = 6, that is, when x = 3. So the crossing is (3, 6). Check both: is 6 = 2(3)? Yes, since 2(3) = 6. And is 6 = 6? Yes. Solution: (3, 6).
7.1.w3 Example 3. Solve y = x + 2 and y = −x + 4. Build a small table and look for the matching row. At x = 1, the first equation gives 1 + 2 = 3, and the second gives −1 + 4 = 3. They agree, so the lines cross at (1, 3). Check both: 3 = 1 + 2, and 3 = −1 + 4. Both true. Solution: (1, 3).
7.1.w4 Example 4, the limitation, on purpose. Solve y = 2x and y = x + 1 but suppose the answer weren't clean. Here it is clean: they meet at (1, 2), since 2 = 2(1) and 2 = 1 + 1, both true. But picture a different system whose lines crossed at, say, (2.5, −1.3). Could you read those numbers off a graph you drew by hand? You couldn't. The best you'd manage is "somewhere near there." That's the honest limit of graphing, and it's exactly why the next two lessons give you methods that produce the exact answer without any squinting.
That's the trade graphing makes. It shows the meaning of a system clearly, because you can see the solution as a crossing point, but it only reliably reads off whole-number crossings. When the intersection looks like it falls between the gridlines, that's not your cue to guess a decimal. It's your cue to reach for substitution or elimination, the methods in the next two lessons.
Check yourself
Each of these has a short answer below it, so you can check your thinking.
- 7.1.c1 "Without graphing yet, what does it mean, in words, that (2, 3) is the solution of a system? What two things must be true?" (It means the point (2, 3) lies on both lines at once. So putting x = 2, y = 3 into the first equation makes it true, and putting the same pair into the second equation makes it true. Both, not one.)
- 7.1.c2 "Two lines on the same plane never touch. What does that tell you about how many solutions the system has?" (None. If the lines never meet, there's no point that's on both, so no (x, y) satisfies both equations. You'll meet this case head-on in 7.4.)
- 7.1.c3 "If a system's solution were (1.5, 4.2), why would graphing by hand let you down, and what would you reach for instead?" (You can't read 1.5 and 4.2 off a hand-drawn grid with any confidence. The point falls between the lines you can draw. You'd switch to an algebraic method, substitution or elimination, to get the exact values.)
(solve by graphing / finding where the two lines meet; give the ordered pair and check both)
Set A: one line is horizontal or through the origin
Reveal answerHide to problem 1
(4, 4) — 4 = 4, 4 = 4Reveal answerHide to problem 2
(3, 6) — 6 = 2(3), 6 = 6Reveal answerHide to problem 3
(3, 3) — 3 = -3 + 6, 3 = 3Set B: two slanted lines