Systems of Equation Fair Race.
Hello, this question relates to Solving Systems of Equations. I was asked to complete a portfolio, here are the directions for said portfolio:
1. Choose two animals with different speeds. You can choose from the chart
that starts at the bottom of this page or do research to choose your own.
2. Design a fair race in which the two animals have an equal chance of winning
if they race at their top speed. Here are a few tips for your design:
a. The race is fair if the two animals could finish the race in the same
amount of time.
b. You can give the slower animal a shorter distance to race.
c. Since this is a video game, the race does not need to be realistic—it
can be any length, and the animals can run at a constant speed.
3. Write a system of two linear equations showing the distance each animal can
travel to model the fair race. Be sure to define all variables.
4. Graph the system to prove that the two animals have an equal chance of
winning the race. Explain how the graph proves the race is fair.
I chose an ostrich with a speed of 40 mph. and a elephant with a speed of 25 mph. I've determined that the equations are as such:
y=40x
y=25x+1.6
And I know that I need to find where the lines of the equations meet on the graph, I just need help solving it, with an explanation so I can better understand it. If anyone would be willing to help me I would be grateful, thank you.
Since the Left hand sides are equal you can set the right hand sides equal. This is The Comparison method and it is a subset of the Substitution method.
That is, you are subbing 40x from the first equation into the second equation.
40x = 25x + 1.6
and solve for x : )
once you have the value of x, then sub it back into the first equation and solve for y
Thank you, this helped me a lot.
To find where the lines of the equations meet on the graph, you can solve the system of equations by setting them equal to each other and solving for the variable.
Let's solve the system of equations:
1. y = 40x
2. y = 25x + 1.6
To find the point of intersection, we need to set the y-values equal to each other:
40x = 25x + 1.6
Now, we can solve for x:
40x - 25x = 1.6
15x = 1.6
x = 1.6/15
x ≈ 0.1067
Substitute this value of x back into one of the original equations (let's use equation 1):
y = 40(0.1067)
y ≈ 4.27
Therefore, the point of intersection for the two lines is approximately (0.1067, 4.27). This means that if both animals run at their top speed, the ostrich and the elephant will meet at this point during the race.
To graphically prove that the race is fair, plot the two equations on a graph. The x-axis represents the time and the y-axis represents the distance traveled.
Plot the graph for equation 1: y = 40x
Start with the point (0, 0) and plot additional points using different values of x. Connect the points to form a straight line.
Plot the graph for equation 2: y = 25x + 1.6
Using the same process, start with the point (0, 1.6) and plot additional points using different values of x. Connect the points to form another straight line.
The point of intersection (0.1067, 4.27) should be on both lines. This shows that at this particular time (x-value), both animals have traveled the same distance (y-value). Therefore, the race is fair, and both animals have an equal chance of winning.
You can also calculate the time it takes for both animals to reach this point by substituting the x-value (0.1067) into one of the equations. For example, using equation 1:
y = 40(0.1067)
y ≈ 4.27
This means that both the ostrich and the elephant will meet at approximately 4.27 units of distance traveled.
To find where the lines of the equations meet on the graph, you need to solve the system of equations. In this case, the system consists of two linear equations:
1. y = 40x
2. y = 25x + 1.6
To solve this system, you can use the method of substitution or the method of elimination. Let's use the method of substitution:
Start by setting the right-hand sides of the equations equal to each other:
40x = 25x + 1.6
To isolate the variable x, you can subtract 25x from both sides:
40x - 25x = 1.6
Combining like terms gives:
15x = 1.6
Now, divide both sides of the equation by 15 to solve for x:
x = 1.6 / 15
Simplifying the division gives:
x ≈ 0.107
Now, substitute this value of x back into either of the original equations to solve for y. Let's use the first equation:
y = 40 * 0.107
Calculating this gives:
y ≈ 4.28
So the solution to the system of equations is approximately:
x ≈ 0.107
y ≈ 4.28
The lines of the equations intersect at the point (0.107, 4.28) on the graph. This means that if both animals race at their top speeds, the ostrich and the elephant will meet at this point. Since they meet at the same time, they have an equal chance of winning the race.