Systems of Equations & World Problems

[AngelShare]

Question: The weights of two beluga whales and three orca whales totals 36,000 pounds. The weights of one of the belugas and one of the orcas add up to 13,000 pounds. Using this information only, how much do the belugas weigh? How much do the orcas weigh?

Part 1: Write a system of equations for this problem and solve it by graphing. (You can write 36,000 and 13,000 as 36 and 13 respectively as long as you remember to put the thousand back on in your answer.) You can use the Graphmatica program to create your graphs, or just draw them on paper and fax or mail them to your instructor. Make sure and indicate the intersection on the graph by writing the ordered pair.

Part 2: Using the same system of equations, solve it by substitution. Be sure to show all work in this process as a final answer is not enough to earn credit. Answer the question in complete sentences.

Part 3: Solve the same system of equations by the addition method. Again, be sure to show all work and indicate your answers in complete sentences.

Before reading the questions properly, I had written down something like "2b + 3o = 36,000" and "b + o = 13,000" but "solve it by graphing" (Part 1) threw me off. Would that, what I had down, be right? If so, how do you graph such a thing?

 

[HallsofIvy]

Question: The weights of two beluga whales and three orca whales totals 36,000 pounds. The weights of one of the belugas and one of the orcas add up to 13,000 pounds. Using this information only, how much do the belugas weigh? How much do the orcas weigh?
Part 1: Write a system of equations for this problem and solve it by graphing. (You can write 36,000 and 13,000 as 36 and 13 respectively as long as you remember to put the thousand back on in your answer.) You can use the Graphmatica program to create your graphs, or just draw them on paper and fax or mail them to your instructor. Make sure and indicate the intersection on the graph by writing the ordered pair.
Part 2: Using the same system of equations, solve it by substitution. Be sure to show all work in this process as a final answer is not enough to earn credit. Answer the question in complete sentences.
Part 3: Solve the same system of equations by the addition method. Again, be sure to show all work and indicate your answers in complete sentences.
Before reading the questions properly, I had written down something like "2b + 3o = 36,000" and "b + o = 13,000" but "solve it by graphing" (Part 1) threw me off. Would that, what I had down, be right? If so, how do you graph such a thing?

Since the problem says "You can write 36,000 and 13,000 as 36 and 13 respectively as long as you remember to put the thousand back on in your answer", it might be simpler to write them as b+ 3o= 36 and b+ o= 13 but what you have is fine.

Do you know that the graphs of equations like these ("linear" equations) are straight lines? Make a "guess" for b and o- they don't have to be values that actually satisfy both equations- b= 0 and o= 0 will work nicely. If you take b= 0 and put it into the equation b+ 3o= 36, you get the equation 3o= 36 so o= 12. Where is the point (0,12) on your graph paper? If you take o= 0 and put it into the equation b+ 3o= 36, you get the equation b= 36. Where is the point (36, 0) on your graph paper? The graph of the equation b+ 3o= 36 is the straight line through those two points.

Now do the same thing with the other equation: when b= 0, what value of o satifies that equation? Where is that point on your graph paper? When o= 0, what value of b satisfies that equation? Where is that point on your graph paper? Draw the straight line through those points.

Every point on one line gives b and o values that satisfy one equation. every point on the other line gives b and o values that satisfy the other equation. The point where the two lines intersect lies on both lines and so satisfies both equations.

 

[Architeuthis Dux]

Yes, your initial equations are correct. To graph them, you would plot points on a coordinate plane where the x-axis represents the number of beluga whales and the y-axis represents the number of orca whales. For example, for the first equation, you could plot the point (2,3) which would represent 2 beluga whales and 3 orca whales, and this point would fall on the line 2b + 3o = 36,000. Similarly, for the second equation, you could plot the point (1,1) which would represent 1 beluga whale and 1 orca whale, and this point would fall on the line b + o = 13,000. By plotting more points and connecting them, you can graph both lines and find the intersection point, which would represent the solution to the system of equations.

Part 2: To solve this system of equations by substitution, you would first solve one of the equations for one of the variables. For example, you could solve the second equation for b by subtracting o from both sides, giving you b = 13,000 - o. Then, you would substitute this expression for b into the first equation, giving you 2(13,000 - o) + 3o = 36,000. From here, you can solve for o and then use that value to find the weight of the beluga whales.

Part 3: To solve this system of equations by the addition method, you would first manipulate the equations to have the same number of one of the variables. For example, you could multiply the second equation by 2, giving you 2b + 2o = 26,000. Then, you would add this equation to the first equation, eliminating the b variable and giving you 5o = 62,000. From here, you can solve for o and then use that value to find the weight of the beluga whales.