Yesterday Lucy walked 2 hours and jogged 1/2 hour and covered 6.25 miles. Today she walked for 3 hours and jogged for 1 hour and covered 10.25 miles. Assuming a constant walking rate and a constant jogging rate, how fast did she walk and how fast did she jog? Define two variables, write a system of equations, and solve to find the walking rate and the jogging rate. Justify your answer by showing how you solved the problem.
w- walking j- jogging
2w + .5j= 6.25 --> 3(2w+.5j)= 3(6.25)
3w + 1j= 10.25 --> -2(3w+1j)= -2(10.25
6w + 1.5j= 18.75>> -.5j=-1.75
-6w - 2j= -20.50 >> j=.3
2w + .5(.3)= 6.25
2w + 15= 6.25
2w= 6.10 w= 3.05
check:
2(3.05) + .5(.3)= 6.25 (correct)
3(3.05) + 1(.3)= 9.45(wrong!!) it should be 10.25.
Up to this line, things are OK.
-6w - 2j= -20.50 >> j=.35
Could you continue and check?
To find the walking rate and the jogging rate, we can define two variables: w for walking rate and j for jogging rate.
Next, we can set up a system of equations using the given information from yesterday and today:
Yesterday:
2w + 0.5j = 6.25
Today:
3w + 1j = 10.25
To solve this system of equations, we can use the method of substitution or elimination. In this case, we will use elimination.
First, we will multiply the first equation by 3 and the second equation by 2 to eliminate the w term:
6w + 1.5j = 18.75
6w + 2j = 20.50
Next, we will subtract the first equation from the second equation:
6w + 2j - (6w + 1.5j) = 20.50 - 18.75
6w + 2j - 6w - 1.5j = 1.75
Simplifying, we get:
0.5j = 1.75
To isolate j, we can divide both sides of the equation by 0.5:
j = 1.75 / 0.5
j = 3.5
Now, we can substitute the value of j back into either of the original equations to find the value of w. Let's substitute it into the first equation:
2w + 0.5(3.5) = 6.25
2w + 1.75 = 6.25
2w = 6.25 - 1.75
2w = 4.5
Dividing both sides by 2, we get:
w = 4.5 / 2
w = 2.25
Therefore, the walking rate is 2.25 miles per hour and the jogging rate is 3.5 miles per hour.
To check our answer, we can substitute these values back into the second equation:
3(2.25) + 1(3.5) = 10.25
6.75 + 3.5 = 10.25
10.25 = 10.25
The equation is correct, so our answer is justified.