Question

two groups of workers are painting a bridge in the bay. the first group is responsible for painting the north side of the bridge, and the second group is responsible for painting the south side of the bridge. the first group has already painted 3 kilometers of the bridge and is painting 2 additional kilometers per day. the second group has already painted 5 kilometers of the bridge and is painting 1 additional kilometer per day. when the two groups have painted the same amount of the bridge, how much of the bridge will they each have painted? write a system of equations, graph them, and type the solution.

Explanation:

Step1: Define Variables

Let \( x \) be the number of days, and \( y \) be the total kilometers painted.

Step2: Write Equations

For the first group: Initial = 3 km, Rate = 2 km/day. So \( y = 2x + 3 \).
For the second group: Initial = 5 km, Rate = 1 km/day. So \( y = x + 5 \).

Step3: Solve the System

Set \( 2x + 3 = x + 5 \).
Subtract \( x \) from both sides: \( x + 3 = 5 \).
Subtract 3: \( x = 2 \) days.

Substitute \( x = 2 \) into \( y = 2x + 3 \): \( y = 2(2) + 3 = 7 \) km.

(Graphing: Plot \( y = 2x + 3 \) (y-intercept 3, slope 2) and \( y = x + 5 \) (y-intercept 5, slope 1). Intersection at (2,7).)

Answer:

When the two groups have painted the same amount, each will have painted 7 kilometers of the bridge.