Question

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 and Equations

Let \( x \) be the number of days, and \( y \) be the total kilometers painted.
For the first group: They start at 0 (assuming no initial painting) and paint 2 km/day, so \( y = 2x \).
For the second group: They already painted 5 km and paint 1 km/day, so \( y = x + 5 \).

Step2: Solve the System

Set the equations equal: \( 2x = x + 5 \).
Subtract \( x \) from both sides: \( x = 5 \).
Substitute \( x = 5 \) into \( y = 2x \): \( y = 2(5) = 10 \).

Step3: Graph Interpretation

The intersection point of \( y = 2x \) (a line through (0,0) with slope 2) and \( y = x + 5 \) (a line through (0,5) with slope 1) is at \( (5, 10) \), confirming the solution.

Answer:

When the two groups have painted the same amount, each has painted 10 kilometers of the bridge.