Question

5 deandre and his sister asha make origami cranes. their goal is to complete 1,000 cranes by the end of the summer. - deandre already has 30 cranes and makes 5 more each day. - asha already has 10 cranes and makes 15 more each day. the graph shows how many cranes, c, each person has made after d days. a. what does the graph show to be the solution of the system? b. what does the solution mean in this context?

Explanation:

Part a

Step1: Recall solution of system from graph

The solution of a system of linear equations from a graph is the point of intersection of the two lines.
Looking at the graph, the two lines (representing DeAndre and Asha's crane - making progress) intersect at a point. From the graph, we can see that the x - coordinate (days, \(d\)) and y - coordinate (cranes, \(c\)) of the intersection point.
We can also calculate it algebraically to verify. Let's set up the equations:
For DeAndre: \(c = 30+5d\)
For Asha: \(c = 10 + 15d\)
Set them equal: \(30 + 5d=10+15d\)

Step2: Solve for \(d\)

Subtract \(5d\) from both sides: \(30=10 + 10d\)
Subtract 10 from both sides: \(20 = 10d\)
Divide both sides by 10: \(d = 2\)

Step3: Solve for \(c\)

Substitute \(d = 2\) into DeAndre's equation: \(c=30 + 5\times2=30 + 10 = 40\)
Or substitute into Asha's equation: \(c = 10+15\times2=10 + 30 = 40\)
So the solution of the system from the graph (and algebra) is when \(d = 2\) and \(c = 40\), or the point \((2,40)\)

The solution \((d = 2,c = 40)\) means that after 2 days of making origami cranes, both DeAndre and Asha will have the same number of cranes, and that number is 40 cranes. DeAndre starts with 30 cranes and makes 5 per day, so after 2 days he has \(30+5\times2 = 40\) cranes. Asha starts with 10 cranes and makes 15 per day, so after 2 days she has \(10 + 15\times2=40\) cranes. So at \(d = 2\) days, their number of cranes (\(c\)) is equal to 40.

Answer:

The solution of the system is \((d = 2,c = 40)\) or when 2 days have passed, both have 40 cranes. In boxed form (for the point) \(\boxed{(2, 40)}\)

Part b