Parallel lines and the meeting point of two lines

On the city map there are two parallel streets: $\displaystyle {y}={2x+1}$ and $\displaystyle {y}={2x-5}$ (the dashed one).

• The two streets have the same slope, 2: for every unit east both climb 2 units north.
• Their $\displaystyle b$ differs, 1 and $\displaystyle -5$: they cut the $\displaystyle y$-axis at different points.
• So the distance between them does not change, and they never meet.

Lines with the same slope and a different $\displaystyle b$ are parallel lines.

The council is building a new street, parallel to the street $\displaystyle {y}={2x+1}$, passing through the car park $\displaystyle P(3,1)$.

The solution below takes the slope from the existing street, and finds $\displaystyle b$ by substituting $\displaystyle P$:

• The streets are parallel, so the new street’s slope is 2: $\displaystyle {y}={2x+b}$.
• We substitute $\displaystyle P(3,1)$: $\displaystyle {1}={2\cdot3+b}$, so $\displaystyle {b}={-5}$.
• The new street’s equation: $\displaystyle {y}={2x-5}$ – the dashed line on the graph above.

The road $\displaystyle {y}={2x+1}$ and the road $\displaystyle {y}={-x+10}$ (the dashed one) meet at the junction $\displaystyle K$.
The meeting point of two lines is the point on both of them: its coordinates satisfy both equations.

The graph shows the junction roughly; the system of equations of the two lines gives its exact coordinates:

• At $\displaystyle K$ the $\displaystyle y$ of the two roads is the same: $\displaystyle {2x+1}={-x+10}$.
• We add $\displaystyle x$ and subtract 1 on both sides: $\displaystyle {3x}={9}$, so $\displaystyle {x}={3}$.
• We substitute in one of the equations: $\displaystyle {y}={2\cdot3+1}={7}$.
• The junction: $\displaystyle K(3,7)$.

A system of two parallel lines has no solution.
For the two streets $\displaystyle {y}={2x+1}$ and $\displaystyle {y}={2x-5}$:

• $\displaystyle {2x+1}={2x-5}$.
• We subtract $\displaystyle 2x$ on both sides: $\displaystyle {1}={-5}$ – an equality that is never true.
• No $\displaystyle x$ satisfies the equation, so the streets have no junction.

In economics too: two price plans with the same price per unit and different fixed costs are parallel lines – one of them is always cheaper, by the same amount.

A town has a straight street whose equation is $\displaystyle {y}={0.5x+2}$.
The council wants to build a street parallel to it, passing through the shopping centre $\displaystyle M(4,-1)$.

Write the new street’s equation on the graph board.

A straight cycle path passes through the points $\displaystyle (0,5)$ and $\displaystyle (2,6)$.
Is the path parallel to the street $\displaystyle {y}={0.5x+2}$?
Explain with a calculation.

Two roads: $\displaystyle {y}={x+1}$, and $\displaystyle {y}={-2x+7}$ (the dashed one).

Mark the junction of the two roads on the graph.

Solve the system of equations, and find the exact coordinates of the junction.

A delivery van drives in a straight line through the points $\displaystyle A(0,1)$ and $\displaystyle B(2,5)$.
A second van drives in a straight line through the points $\displaystyle C(0,10)$ and $\displaystyle D(2,8)$.

Write the equation of each of the routes on the graph board.

Write the system of equations of the two routes and solve it.

At which point do the two vans’ routes cross?
If the two delivery firms want to pass parcels from van to van, where should they meet?

Two removals firms charge by distance: $\displaystyle x$ is the distance in kilometres, and $\displaystyle y$ the price in shekels.
In firm A the price is $\displaystyle {y}={3x+20}$, and in firm B – $\displaystyle {y}={3x+35}$.
Solve the system: at what distance do the two firms charge the same price?
Which firm is cheaper, and by how much?