Same Math › Grade 8 › Solving systems by graphing
What is solving a system by graphing?
A system is two equations that are both true at once. Graph both lines: the point where they cross works in both equations, so it's the solution. For y = x + 1 and y = −x + 3, the lines cross at (1, 2), so x = 1 and y = 2.
- Each line shows every point that works for one equation.
- Two lines means two conditions at once.
- The crossing point works for both: the solution.
Why schools teach it
It shows what a solution means
Eighth graders learn that the solution to a system is the point where the graphs intersect, and solve systems by graphing, by inspection and with algebra (8.EE.C.8). Graphing comes first because it makes the meaning visible.
It also explains odd cases: parallel lines never cross, so there's no solution, and lines that sit on top of each other have infinitely many.
At the kitchen table
What to say, and what to skip
Say this
Where does each line cross the y-axis?
That's b in y = mx + b.Use the slope to find more points.
Where do they cross?
Check that point in both equations.
Skip this
- Graphing sloppily.A ruler and grid paper matter: a small slip moves the crossing point.
- Stopping at x.The solution is a point: both x and y.
- Skipping the check.Substitute into both equations.
See it in action
Solving y = 2x and y = x + 3
Graph y = 2x: through (0, 0), up 2 for each 1 across
Graph y = x + 3: start at (0, 3), up 1 for each 1 across
Find the crossing and check
(3, 6): 6 = 2(3) and 6 = 3 + 3
Real problems: two phone plans, and when do they cost the same? The crossing point answers it.
Free practice sheet
Make a systems by graphing sheet
Loading practice sheet…
Fridge cheat sheets
Math Fridge Cheat Sheets
Every K–8 math method in one printable book (159 pages): a cheat sheet page like this guide, plus a practice sheet and its answer key for each, in Classic and handwritten Notebook styles. $9 one time, or all 7 subjects for $29.99. See all prices.
Teacher? This method has a classroom pack on TeachersPayTeachers: an anchor chart, a parent letter and practice sheets with answer keys, $4. See the pack
Parents: tell your child's teacher about this guide
Parents also ask
What if the lines don't cross?
Parallel lines (same slope, different starting point) have no solution.
What if they're the same line?
Every point on it works: infinitely many solutions.
Is graphing the only way?
No. Substitution and elimination come next, and work when the crossing isn't on a grid corner.
What's y = mx + b?
Slope-intercept form: m is the slope and b is where the line crosses the y-axis.