Let's tackle the question in a structured manner as per the parts given.
Part A: Write a pair of linear equations
Jacob spends a total of 60 minutes in the gym every day, doing two activities: freehand exercises and running on the treadmill. We are given that he runs on the treadmill for 30 minutes longer than he does freehand exercises.
Let [tex]\( y \)[/tex] represent the number of minutes Jacob spends doing freehand exercises. Let [tex]\( x \)[/tex] represent the number of minutes Jacob spends running on the treadmill.
We are given two pieces of information to create our system of equations:
1. The total time spent in the gym is 60 minutes: [tex]\( x + y = 60 \)[/tex].
2. Jacob runs on the treadmill for 30 minutes longer than he does freehand exercises: [tex]\( x = y + 30 \)[/tex].
So the pair of linear equations is:
[tex]\[ x + y = 60 \][/tex] (Total time equation)
[tex]\[ x = y + 30 \][/tex] (Relationship between treadmill and freehand time)
Part B: How much time does Jacob spend doing freehand exercises?
To find the time Jacob spends doing freehand exercises, we need to use the two equations to solve for [tex]\( y \)[/tex].
Starting with the relationship equation:
[tex]\[ x = y + 30 \][/tex]
Now we will substitute this expression for [tex]\( x \)[/tex] into the total time equation to find [tex]\( y \)[/tex]:
[tex]\[ (y + 30) + y = 60 \][/tex]
[tex]\[ 2y + 30 = 60 \][/tex]
Subtract 30 from both sides:
[tex]\[ 2y = 30 \][/tex]
Now divide by 2:
[tex]\[ y = 15 \][/tex]
Jacob spends 15 minutes doing freehand exercises.
Part C: Is it possible for Jacob to have spent 40 minutes running on the treadmill?
The relationship between the time spent on the treadmill and the time spent doing freehand exercises is defined by the equation:
[tex]\[ x = y + 30 \][/tex]
We need to check if Jacob can spend 40 minutes on the treadmill, we can substitute [tex]\( x \)[/tex] with 40 in the relationship equation and see if the total time adds up to 60 minutes.
Substitute [tex]\( x \)[/tex] with 40 in the relationship equation:
[tex]\[ 40 = y + 30 \][/tex]
Subtract 30 from both sides:
[tex]\[ 10 = y \][/tex]
This implies that Jacob would spend 10 minutes doing freehand exercises. Let's now check the total time this would take:
[tex]\[ x + y = 40 + 10 = 50 \][/tex]
But since we know that Jacob spends a total of 60 minutes at the gym, the sum of treadmill time and freehand time must equal 60 minutes. Since 50 does not equal 60, it is not possible for Jacob to spend 40 minutes on the treadmill if he also wants to maintain the 30-minute longer treadmill time than freehand exercise time within the 60-minute total gym time constraint.
So, the conclusion is no, it's not possible for Jacob to spend 40 minutes on the treadmill given the conditions specified.
