Systems of Equations: Three Methods, One Intersection Point
A system of equations asks where two lines cross. Three ways to find that point — graphing, substitution, elimination — and how to recognize when there are zero, one, or infinite solutions.
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Where do two lines meet?
A system of equations is just two equations sharing the same variables. The solution is the point (x, y) that satisfies both equations — visually, the point where the two lines cross.
Big idea
One linear equation describes infinite points (a whole line). Two linear equations together usually pin down one specific point — the intersection.
Three possible outcomes
Different slopes → cross once. Same slope, different intercept → parallel, no solution. Same line entirely → infinite.
What does the solution represent?
A solution to a system of two linear equations is the point where:
The solution to a system is where both equations are satisfied simultaneously, which is where the lines intersect.
No solution
A system of equations has no solution. The lines are:
📌 No solution = lines never intersect = parallel lines. Parallel = same slope, different y-intercepts.
Infinite solutions
Infinitely many solutions means the two equations represent:
📌 Same line = same slope AND same y-intercept. Every point on the line is a solution.
Method 1: Substitution
Best when one equation already has a variable isolated (like “y = ...”).
Best when both equations are in standard form (Ax + By = C). Add or subtract to eliminate one variable.
2x + 3y = 162x − y = 4 (subtract to eliminate x)4y = 12 → y = 32x − 3 = 4 → x = 3.5Solution: (3.5, 3)
Method 3: Graphing
Plot both lines and read the intersection. Best for visual confirmation, but coordinates that aren't integers are hard to read off a graph.
3-second recap
The solution is the point (x, y) that makes both equations true.
Different slopes → one solution. Same slope, different intercept → no solution. Identical line → infinite.
One variable already isolated → substitution. Both in standard form → elimination.
Check yourself
Quick check #1
What does it mean if two equations in a system give the same line when graphed?
If the two equations describe the SAME line, every point on that line satisfies both — infinitely many solutions. Parallel lines (same slope, different intercept) give no solution; intersecting lines give one solution.
Quick check #2
Solve the system: y = 2x + 1 and y = 3x − 4.
Substitution: 2x + 1 = 3x − 4. Subtract 2x: 1 = x − 4. So x = 5, then y = 2(5) + 1 = 11. Solution: (5, 11).