Two courses, one meeting point

Erklärus drew two flight paths as straight lines in a coordinate system – they crossed at exactly one point. “Two ships that are supposed to meet somewhere always needed exactly this calculation. Do you remember?”

Liri: “Yes! You had to solve both courses at the same time to find the meeting point.”
Erklärus: “That is exactly a system of linear equations. Let’s practise the methods for it.”

📐 The graphical solution – the intersection

If you draw both equations as straight lines in a coordinate system, their intersection (x, y) is the solution of the system.

Solution (x, y)
Erklärus: “The intersection is the only point that lies on BOTH lines at the same time – that is exactly what it means to satisfy both equations at once.”

🔄 The equating method

Solve both equations for the same variable and set the expressions equal to each other.

📐 Example: y=2x+1 and y=x+3

Both solved for y, so set them equal: 2x+1 = x+3
x = 2, substituted: y = 2×2+1 = 5
Solution: (2, 5)

➡️ The substitution method

Solve one equation for one variable and substitute this expression into the other equation.

📐 Example: y=x+1 and 2x+y=7

Substitute y=x+1 into the second equation: 2x+(x+1)=7
3x+1=7 → x=2, substituted: y=2+1=3
Solution: (2, 3)

➕ The elimination (addition) method

Multiply the equations so that one variable drops out when you add.

📐 Example: x+y=5 and x−y=1

Add both equations: 2x = 6 → x=3
substituted into x+y=5: y=2
Solution: (3, 2)

x+y=10 and x−y=2 – how big is x? (elimination method)

🧮 Liri’s simultaneous-equations calculator – interactive

Enter two lines of the form y=mx+b – the intersection appears automatically:

📈 Intersection of two lines
Line 1: m₁= b₁=
Line 2: m₂= b₂=
🌌 Simultaneous equations in the universe

Mission control centres constantly solve systems of equations to work out when and where two flight paths meet – for example when two spaceships dock.

❌ Common mistake

If two lines have the same gradient, they are parallel – then there is no solution (no intersection)!

📝 Exercise 1
y=x+2 and y=3x−2 – how big is x at the intersection? (equating method)
📝 Exercise 2
x+y=8 and x−y=4 – how big is x? (elimination method)
📝 Exercise 3
y=2x and x+y=9 – how big is x? (substitution method)

The meeting point found

Liri now worked out the exact meeting point of two courses and wrote it onto her star map.

Liri: “Three different ways, but all of them lead to the same meeting point. Now I can cross every route with another one.”
“Solving simultaneous equations: equating, substitution or elimination method. Graphically: the intersection of two lines. Same gradient = no solution.”
What you learned today
Intersection (x, y)
Equating · substituting · eliminating – three ways, the same solution
🌟

Chapter 49 mastered!

Liri can now solve systems of linear equations with three methods.

Erklärus’s notebook – page 49:
“Equating, substituting, eliminating – all three lead to the same intersection.”
← Back to overview 49 – Simultaneous Equations