Draw two lines, find the shared dot
A point on both lines solves the pair.
One equation, one line
The equation x + 3y = 6 has many solutions: (0, 2), (3, 1), (6, 0), (−3, 3)… They all sit on one straight line.
How many of its solutions do you need to draw the line?
What this lesson covers
The idea
Each equation of the pair is represented by a line, drawn through two of its solutions; a point common to both lines gives a solution of the pair of equations.
One equation, one line
The equation x + 3y = 6 has many solutions: (0, 2), (3, 1), (6, 0), (−3, 3)… They all sit on one straight line.
How many of its solutions do you need to draw the line?
- One
- Two
- All of them
Draw both lines
For each equation, tap two (or more) values of x. The machine finds y, and the line is drawn through your dots. Then see where the two lines meet.
The shared point
Each linear equation is drawn as a line, through two of its solutions. A point that lies on both lines is a solution of the pair.
Here the lines meet at (6, 0), so x = 6 and y = 0. Check: 6 + 3 × 0 = 6, and 2 × 6 − 3 × 0 = 12. Both are true.
Why does this work? A point on line (1) makes equation (1) true. A point on line (2) makes equation (2) true. A point on both lines makes both true.
Notes
Draw each equation as a line through two of its solutions. A point on both lines is a solution of the pair.
Check yourself
Which two points both lie on the line 2x + y = 6?
The lines of x + y = 7 and x − y = 3 meet where x = 5. What is y there?
Answer: 2
5 + y = 7 gives y = 2. Check in the other equation: 5 − 2 = 3 ✓. The lines meet at (5, 2).
Champa buys x pants and y skirts. Her equations are y = 2x − 2 and y = 4x − 4. Their lines meet at (1, 0). What did she buy?
A point lies on line (1) but not on line (2). Is it a solution of the pair?
- (0, 6) and (3, 0) — correct. Yes! 2 × 0 + 6 = 6, and 2 × 3 + 0 = 6.
- (0, 3) and (6, 0). (0, 3) fails: 2 × 0 + 3 = 3, not 6.
- (0, 6) and (6, 0). (6, 0) fails: 2 × 6 + 0 = 12, not 6.
- (1, 3) and (2, 4). (1, 3) fails: 2 × 1 + 3 = 5, not 6.
- 1 pant and no skirt — correct. Yes! x = 1 pant and y = 0 skirts. Both equations hold: 2 × 1 − 2 = 0 and 4 × 1 − 4 = 0.
- 1 skirt and no pant. x counts pants and y counts skirts, so (1, 0) means 1 pant.
- Nothing: the lines do not meet. They do meet, at (1, 0). That point is the solution.
- No: it makes only equation (1) true — correct. Yes! A solution of the pair must make both equations true.
- Yes: one equation is enough. A solution of the pair must satisfy both equations.
- Yes: every point of a line is a solution. Every point of line (1) solves equation (1) only. The pair needs both.