Solving Systems of Equations by Graphing - Problem 3
Use the information from the word problem to set up two equations that model the problem. Make sure you properly identify which is the independent variable and which is the dependent variable. Next, graph the two lines by first plotting the y-intercept (the constant) and using the slope (the coefficient of the independent variable) to plot a second point. Once you have graphed the two lines, identify the point where the lines intersect. This point is the solution to the system of equations. Check your answer by plugging in the x and y value to the two equations. If the equations are true, then you have found the correct solution.
As you guys know from doing your Math homework, sometimes you get like really boring word problem, and this is an example because it’s about tomato plants. And you guys to be honest I personally I’m not a tomato plant grower, but this is still a type of problem you could solve with something that’s more relevant to your life. Let’s try it.
You buy two tomato plants and want to predict when you can expect them to reach the same height. Plant A is 6 inches when you buy it, and it grows half an inch per week. I’m going to pause there and start writing some stuff down because this is like a super long word problem, and I don’t want to forget what’s going on in my head. Plant A is 6 inches when I buy it, and it grows half an inch per week. I chose to use y for my dependent variable to represent height because how tall it is depends on how many weeks have gone by.
Okay plant b is 3 inches when you buy it, and it grows 1 inch per week, you can write 1x or just write x that means there’s an implied one there. We have a coefficient, like a secret coefficient of 1; you don’t have to write it. Okay write and graph a system of equations to predict when the plants will reach the same height. Part A is done. Here is my system of equations, it represents the two different plants and their heights where y is height, and x is time in weeks.
Let’s go over here and start graphing them. When I set up my graph, I only did it in quadrant 1 because I know I’m not going to have any negative values, plants don’t grow in negative amounts and I can’t think about time in negative world unless there was like time in the past I guess, but I chose to do all positive values for my graph.
Okay my plant A is going to be in blue. It starts at 6 for the y intercept, and goes up one box and then over two. It starts at 6, 1, 2, 3, 4, 5, 6 goes up 1 over 2, that means it’s growing half an inch every week. I’m going to draw a few different points so that I can make sure my line is straight when I stick my ruler on there. Okay good, there is plant A I’m going to label it. It really helps with problems like this if you can label everything and maybe use colour if you’re a visual person like me.
Okay plant B starts at 3 inches and goes up 1 inch per week, so the y intercept here is 3, 1, 2, 3 going up 1 over 1. Using my ruler I’m going to draw these dots, but first I want using my ruler I’m going to draw the line, but first I’ve put on all kinds of dots because I can already tell where my solution point is going to be. Draw your line and make sure you label it for point B, excuse me plant B, plant B okay. So the solution is the place where the lines cross. In our situation if you look at the graph and if you did your graph correctly, your solution will be at 6 week 1, 2, 3, 4, 5, 6 weeks and then if I come over here to how tall the plants are it’s 1, 2, 3, 4, 5, 6, 7, 9 inches okay so my point is (6,9).
When the question asked me to say how long it will take before the graphs reach the same height, my answer is going to be in x value because x represents time or when in the problem. So when it says when are they going to reach the same height? My answer is going to be 6 weeks. After 6 weeks, both plants are 9 inches tall.
Another cool thing about this graph is you can see how now plant B is going to be taller than plant a forever more assuming they continue rowing at the same rate. So before I leave you guys, I want to tell you that a lot of my students tend to skip problem like these because to them they are boring, and because they don’t want to read. You guys can do these problem, they’re not too tricky, just give it your best effort.