Carnegie Learning Math Series Volume 14th Edition Chapter 1 Linear Equations
Carnegie Learning Math Series Volume 1 4th Edition Chapter 1 Exercise 1.1 Solution Page 4 Problem 1 Answer
Given the height is 150 feet long and rate is 5 feet per hour.
The aim is to find total feet of walkway will there be after the park rangers work for 5 hours.
For 1 hour walk way can be built till 5 feet.
For 5 hours walkway built is 5+5+5+5+5=25 feet.
Therefore, the distance built for 5 hours is 25 feet.
Page 4 Problem 2 Answer
Given that walkway is 150 feet long and rate is 5 hours per feet.
The aim is to find the total feet of walkway will there be after the park rangers work for 7 hours.
For 1 hour walk way can be built till 5 feet.
For 7 hours walkway built is 7×5=35 feet.
Therefore, the feet built is 35 feet for 7 hours.
Page 4 Problem 3 Answer
Given that 150 feet and the rate is 5 hours per feet.
The aim is to find total number of feet of walkway built, given the amount of time that the rangers will work.
To find the walk way built is, 150=5t
Where t is the time taken to build the walkway.
Therefore, the expression is 150=5t.
Page 4 Problem 4 Answer
Given the distance is 500 feet.
The aim is to find the number of hours.
Using the formula for part (c),500=5t
t=100 hours
Therefore, the time taken to finish the 500 feet is 100 hours.
Page 4 Problem 5 Answer
Given that 150 feet and the rate is 5 hours per hour.
The aim is to find the reasoning for part(d).
Using the formula for part(c) is because the time taken for 5 feet is 1 hour.
So, the time taken for 1 feet is 1/5 hour.
Therefore, for 500 feet then 500/5=100 hours.
Therefore, the number of hours to built the halfway of 500 feet is 100 hours.
Page 4 Problem 6 Answer
Given that 150 feet and rate is 5 feet per hour.
The aim is to find the mathematical operation used in part (d).
The mathematical calculation used is division in part(d).
The mathematical calculation used is division.
Page 4 Problem 7 Answer
Given that 150 feet distance and rate is 5 feet per hour.
The aim is to find the equation and determine the value.
The formula is,
500=5t
t=500/5
t=100
Where t is time.
The formula is 500=5t and the value of t is 100 hours which is true.
Solutions For Linear Equations Exercise 1.1 In Carnegie Learning Math Series Page 4 Problem 8 Answer
Given that walkway is 150 feet and the rate is 5 feet per hour.
The aim is to Interpret your answer in terms of this problem situation.
The problem is when the distance is given and the rate is given.
Using division the number of hours can be found.
Therefore, the number of hours is the ratio of distance of walkway and the rate.
Page 5 Problem 9 Answer
Given that 270 feet.
The aim is to find the number of hours.
The mathematical formula is,
Speed=distance/time
time=270/speed
To determine time we require speed and mathematical operation is division.
Page 5 Problem 10 Answer
Given that 270 feet of walkway.
The aim is to find why using these mathematical operations gives you the correct answer.
The formula is, correct since speed=distance/time is well known result.
Therefore by using the formula the division gives the true value.
Page 5 Problem 11 Answer
Given: 270 feet of path is to be built and the rangers can build the path at a rate of 5 feet per hour.
To find: An equation that can be used to determine the amount of time it will take to complete the 270 feet of walkway
Procedure: We will divide the number of feet of path to be built by the rate of path built per hour.
Let the time taken to complete the work be x
Let the walkway to be built bey=270 feet
Let the rate per hour at which the walkway can be built bes=5 feet
The equation for the time taken will be x=y/s (Using the formula Time taken=Path to be built
Path built per hour)
The equation that can be used to determine the amount of time taken to complete the work is x=y/s.
Page 5 Problem 12 Answer
Given: 270 feet of path is to be built and the rangers can build the path at a rate of 5 feet per hour.
To find: The value of the variable that will make the equation true
Procedure: We will put the values of the known variables to find the value of the unknown variable.
From part (c), the equation was x=y/s
To find the value of the variable, we will put the values of the known variables
Here the value of y=270 and s=5
⇒x=270/5
⇒x=54 hours
Therefore,x=54 will make the equation true.
The value of variable to make the equation true is x=54.
Page 5 Problem 13 Answer
Given:270 feet of path is to be built and the rangers can build the path at a rate of 5 feet per hour.
To find: The answer in terms of this problem situation
Procedure: We will interpret and write the answer using language.
From the part (d), we found that the value ofx=54
The answer to the problem situation is that the rangers will need a total of 54 hours to build 270 feet of walkway
The answer to the problem situation is the rangers will need 54 hours to build the 270 feet walkway at a rate of 5 feet per hour.
Carnegie Learning Math Series 4th Edition Exercise 1.1 Solutions Page 6 Problem 14 Answer
Given: 100 feet of the path is to be built and the rangers can build the path at a rate of 5 feet per hour.
To find: The mathematical operation used to determine the answer.
Procedure: We will use division as the mathematical operation.
Path to be built=100 feet
Path built per hour=5 feet
Time taken=100/5
=20 hours
The rangers will need to work for 20 hours to build the 100 feet walkway.
We have performed division as the mathematical operation to determine the answer using the formula:
Time taken=Path to be built/Path built per hour
The rangers will need to work for 20 hours to build the100 feet walkway.
The mathematical operation performed to determine the answer is division.
Page 6 Problem 15 Answer
Given: Given: 100 feet of path is to be built and the rangers can build the path at a rate of 5 feet per hour.
To find: An equation that can be used to determine the amount of time it will take to complete the 100 feet of walkway.
Procedure: We will divide the number of feet of path to be built by the rate of path built per hour.
Let the time taken to complete the work be x
Let the walkway to be built bey=270 feet
Let the rate at which the walkway can be built be s=5 feet per hour
The equation for the time taken will be x=y/s (Using the formula Time taken=Path to be built/ Path built per hour)
The equation that can be used to determine the amount of time taken to complete the work is x=y/s
Page 6 Problem 16 Answer
Given: Given: 100 feet of path is to be built and the rangers can build the path at a rate of 5 feet per hour.
To find:
An equation that can be used to determine the amount of time it will take to complete 100 feet of walkway.
If our answer makes sense.
Procedure: We will put the values of the known variables to find the value of the unknown variable.
From part (b). the equation was x=y/s
To find the value of the variable, we will put the values of the known variables
Here the value of y=100 and s=5
⇒x=100/5
⇒x=20 hours
Therefore,x=20 will make the equation true.
Yes, our answer makes sense as it is less than the number of feet of walkway to be built which proves the division.
The value of variable to make the equation true isx=20 hours.
Yes, our answer makes sense as it is less than the number of feet of walkway to be built which proves the division.
Page 6 Problem 17 Answer
Given: 100 feet of path is to be built and the rangers can build the path at a rate of 5 feet per hour.
To find: An equation that can be used to determine the amount of time it will take to complete the 100 feet of walkway.
Procedure: We will interpret and write the answer using language.
From part (c), we found that the value of x=20
The answer to the problem situation is that the rangers will need a total of 20 hours to build the100 feet of walkway.
The answer to the problem situation is the rangers will need 20 hours to build the 100 feet walkway at a rate of 5 feet per hour.
Linear Equations Solutions Chapter 1 Exercise 1.1 Carnegie Learning Math Series Page 7 Problem 18 Answer
Given: The cub is 77 feet below the ground level. The ranger is pulling the rope at the rate of 7 feet per minute.
To find: How many feet below the surface of the ground will the cub be in 6 minutes.
Procedure: We will calculate the distance pulled up by the ranger in 6 minutes and then subtract that distance from/the total distance.
Using the formula Speed=Distance/Time
⇒Distance=Speed×Time
Here the speed=7 feet per minute, and time=6 minutes
So, the distance pulled up by the ranger in 6 minutes:
Distance pulled up=7×6
=42 feet
Remaining distance below the surface of ground=Total distance−Distance pulled up
=77−42
=35 feet
The cub is 35 feet below the surface of ground.
The cub will be 35 feet below the surface of ground in 6 minutes.
Page 7 Problem 19 Answer
Given: The cub is 77 feet below the surface.
The ranger is pulling it up at a rate of 7 feet per minute.
To find: How many feet below the surface of the ground will the cub be in 11 minutes
Procedure: We will calculate the distance pulled up by the ranger in 11minutes and then subtract that distance from/the total distance.
Using the formula Speed=Distance/Time
⇒Distance=Speed×Time
Here, the speed=7 feet per minute, and time=11 minutes
So, the distance pulled up by the ranger in 11 minutes:
Distance pulled up=7×11
=77 feet
Remaining distance below the surface of ground=Total distance−Distance pulled up
=77−77
=0 feet
The cub is 0 feet below the surface of the ground.
That means the cub is pulled up and hence is rescued.
The cub will be 0 feet below the surface of the ground after 11 minutes i.e. it is pulled up.
Page 7 Problem 20 Answer
Given: The cub is 77 feet below the surface. The ranger is pulling it up at a rate of 7 feet per minute.
To find: A variable for the amount of time spent pulling the cub up the ravine.
Write an expression that represents the number of feet below the surface of the ground the cub is.
Procedure: First we will assume a variable for the time spent pulling up the cub up the ravine.
Then we will assume a variable for representing the number of feet below the surface of ground the cub is.
We will write the expression as equal to the total distance below the surface−
the distance by which the cub is pulled up by the ranger.
Let the variable for the amount of time spent pulling the cub up the ravine bet minutes
And the number of feet below the surface of the ground the cub is be h feet
Total distance below the ground surface=77 feet
The speed/rate at which the ranger can pull up the cub=7 feet per minute
Therefore, the distance by which the cub is pulled when the time is given=7t feet
Expression for a number of feet below the surface of the ground the cub is can be written as:
h=77−7t
Variable for the time in minutes spent pulling the cub up the ravine=t.
Expression for a number of feet below the surface of the ground the cub ish=77−7t.
Carnegie Learning Math Series Chapter 1 Exercise 1.1 Free Solutions Page 7 Problem 21 Answer
In the question, the speed at which the cub is being pulled is given and the distance for which we have to calculate the time is given.
use a variable for the time and frame an equation to get the answer.
Frame an equation using the information given.
Let the time taken be x.
It is given that the cub is being pulled at a rate of 7 feet per minute, so speed is 7 feet per minute.
We have to calculate the time taken to pull the cub 14 feet from the surface, so distance is 14.
Distance=speed×time
Therefore, 14=7×x or 7x=14.
Solve the equation to get the answer.
we have, 7x=14
⇒x=2
( Divide both the sides by 7)
We get the value of x=2, so the cub will be 14 feet from the surface in 2 minutes, when it is being pulled at a rate of 7 feet per minute.
The cub will be 14 feet from the surface in 2minutes.
Step-By-Step Solutions For Carnegie Learning Math Series Chapter 1 Exercise 1.1 Page 7 Problem 22 Answer
The speed at which the cub is being pulled is given, we have to find the time taken to pull the cub 14
feet from the surface, so here the distance is also given that is 14 feet, we use a variable x, for the time taken to pull the cub and then frame an equation using the formula of distance speed and time, that is, Distance=speed×time.
As we know the speed and the distance, we can easily calculate the time it will take by putting the speed and distance in the formula.
On solving we get the value of x as 2, we had used x for the time taken, so the time taken to pull the cub 14 feet from the surface is 2 minutes.
Therefore we used the formula of speed distance and time to get the answer.
We used a variable to denote the time taken and then framed an equation using the speed distance time formula to get the answer.
Page 7 Problem 23 Answer
In order to determine our answer in part (d), we used two mathematical operations, first, we framed an equation using the speed distance time formula and then solved the equation we framed.
To solve the equation, we divided both the sides by 7 to remove the multiplication with x.
On doing this, we get the value of x, which is the time taken.
First mathematical operation we used is we frame the equation.
let the time taken be x distance=speed×time
therefore, 14=7x
As given speed is 7 and distance is 14.
The equation is, 7x=14
The second mathematical operation we used is division to remove or undo the multiplication.
We have, 7x=14 therefore we divide both the sides by 7 to remove the multiplication 7 from x.
we get, x=2(on dividing both sides by 7)
The mathematical operations we used are, first we framed an equation and then used division to undo the multiplication.
Carnegie Learning Math Series Exercise 1.1 Student Solutions Page 7 Problem 24 Answer
In order to solve part (d), we framed an equation and then used the mathematical operation of division or inverse operation of Division to undo or remove the multiplication.
Here we first framed an equation using the given information, as we were given the speed and distance , so we used the formula Distance=speed×time, to get the time.
We get an equation of the form, 7x=14.
As we know an inverse operation of division is used to undo multiplication, and in the equation we have to remove 7 (which is multiplied to x), to get the value of x,Therefore we used division to remove the multiplication.
So, using these mathematical operations gives us the correct answer, as we have used the operations according to the requirement of the question and as per the requirement of the question these mathematical operations are best suited.
The mathematical operations give us the correct answer because we used them according to the requirement of the question and these are the easiest and appropriate mathematical operations to be used for this type of question.
Page 7 Problem 25 Answer
Given that h=77−7t.
The aim is to find the number of minutes it takes for the cub to be 14 fee below the surface of the ground by setting the expression.
Given that t=14 then,
h=77−98=−21
Therefore, the time is 21 minutes.
Page 8 Problem 26 Answer
In the question, the speed at which the cub is being pulled is given and the distance for which we have to calculate the time is given. use a variable for the time and frame an equation to get the answer.
Frame an equation using the information given.
Let the time taken be x
It is given that the cub is being pulled at a rate of 7 feet per minute, so the speed here is 7.
The distance for which we have to calculate the time is 28 feet, therefore the distance is 28.
Distance=Speed×time
so, 28=7×x or, 7x=28
Solve the equation.
we have, 7x=28
Use inverse operation of division to undo multiplication.
we get ,x=4(divide both side by 7) therefore in 4 minutes the cub will be 28 feet from the surface.
In 4minutes the cub will be 28 feet from the surface.
Page 8 Problem 27 Answer
In order to determine our answer in part (d), we used two mathematical operations, first we framed an equation using the speed distance time formula and then solved the equation we framed.
To solve the equation by using the inverse operation, division to undo the multiplication.
First operation we use is framing the equation.
As the speed and the distance is given, we assume time taken as x.
therefore using the speed distance time formula the equation becomes,
7x=28, where the distance is 28 and the speed is 7.
After the first step we use inverse operation of division to undo the multiplication.
We have, 7x=28
Divide both side by 7 to remove it.
we get, x=4(divide both side by 7)
We first framed an equation and then used the mathematical operation division to undo the multiplication.
Page 8 Problem 28 Answer
In order to solve the question, we framed an equation and then used the mathematical operation of division or inverse operation of Division to undo or remove the multiplication.
Here we first framed an equation using the given information, as we were given the speed and distance, so we used the formula Distance=speed×time, to get the time.
So we get, 7x=28.
As we know an inverse operation of division is used to undo multiplication, and in the equation we have to remove 7 from x, to get the value of x.
So we used the inverse operation of division.
Therefore, using these mathematical operations gives us the correct answer, as we have used the operations according to the requirement of the question and as per the requirement of the question these mathematical operations are best suited.
The mathematical operations give us the correct answer because we used them according to the requirement of the question and these are the easiest and appropriate mathematical operations to be used for this type of question.
Linear Equations Exercise 1.1 Carnegie Learning 4th Edition Answers Page 8 Problem 29 Answer
The distance and speed are given, use a variable to denote the time taken, and frame an equation using the speed distance time formula. Solve the equation
Frame an equation.
The distance here is 28 feet.
It is given that the cub is pulled at a rate of 7 feet per minute, so speed is 7.
Let the time taken be x.
Distance=Speed×time
Therefore the equation is, 28=7×x or, 7x=28
Determine the value of the variable by solving the equation.
we have, 7x=28
On Dividing both sides by 7,
we get, x=4.
Therefore the value of the variable is 4 that makes the equation true.
The equation that can be used to determine in how many minutes the cub will be 28 feet from surface is, 7x=28.
The value of the variable that makes the equation true is 4.
Page 10 Problem 30 Answer
In the given method, inverse operations are used in each step.
In the first step, we add 6 on both the sides, to undo the subtraction.
In the second step ,we divide both the sides by 2 to undo the multiplication.
In the first step addition is used to undo the subtraction.
In the second step division is used to undo the multiplication.
Page 10 Problem 31 Answer
We are given 2m−6=22
We are required to find the difference between the two given strategies to solve the given equation.
In method 1, the concept of inverse operations is done on both sides i.e., the addition is done to undo the subtraction and then division to undo multiplication.
In method 2, the equation 2m−6=22 is added with another expression i.e., 6=6.
So, the approach is different for the first step and second step is the same to calculate the solution.
There aren’t many differences between both the methods to solve the equation but the approach to the first step is different.
Generally, the variable is isolated from all the constant terms to determine the solution of the equation.
Page 10 Problem 32 Answer
We are given 2m−6=22.
We are required to verify that m=14 is the solution of the equation.
We will put the value in the given equation to verify it.
Given: 2m−6=22
To verify that m=14 is a solution, we will substitute it in the equation.
Considering the left-hand side of the equation, we get
2(14)−6
=28−6
=22
which is equal to the right-hand side of the equation.
Hence, it is verified that m=14 is the solution of the equation
2m−6=22.
It is verified that m=14 is a solution of the equation
2m−6=22 because when we substitute the value of m in the equation, we get a valid equation.
Page 10 Problem 33 Answer
We are given 5v−34=26.
We are required to solve the equation.
We will isolate the variable from the constants to determine the solution of the given equation.
Given: 5v−34=26
We will isolate the variable.
Adding 34 on both sides, we get
5v=60
Dividing both sides by 5, we get
v=12
The solution of the equation 5v−34=26 is v=12.
The steps can be shown as:
5v−34=26
5v=60
v=12
The strategy to solve any two-step equation is to isolate the variable by undoing the operations or using inverse operations.
Page 10 Problem 34 Answer
We are given 3x+7=37
We are required to solve the equation.
We will isolate the variable from the constants to determine the solution of the given equation.
Given is 3x+7=37
Isolating the variable by using inverse operations, we get
Subtracting both sides by 7, we get
3x=30
Dividing both sides by 3, we get x=10
The solution of the equation 3x+7=37 is x=10.
The steps can be shown as:
3x+7=37
3x=30
x=10
The strategy to solve any two-step equation is to isolate the variable by undoing the operations or using inverse operations.
Page 10 Problem 35 Answer
We are given 23+4x=83
We are required to solve the equation.
We will isolate the variable from the constants to determine the solution of the given equation.
Given equation is 23+4x=83
Isolating the variable by using inverse operations, we get
Subtracting 23 on both sides, we get
4x=60
Dividing both sides by 4, we get x=15
The solution of 23+4x=83 is x=15.
The steps can be shown as
23+4x=83
4x=60
x=15
The strategy to solve any two-step equation is to isolate the variable by undoing the operations or using inverse operations.
Page 10 Problem 36 Answer
We are given 2.5c−12=13
We are required to solve the equation.
We will isolate the variable from the constants to determine the solution of the given equation.
Given equation is 2.5c−12=13
Isolating the variable using inverse operations, we get
Adding 12 on both sides, we get
2.5c=25
Dividing both sides by 2.5, we get
c=10
The solution of 2.5c−12=13 is c=10
The steps can be shown as
2.5c−12=13
2.5c=25
c=10
The strategy to solve any two-step equation is to isolate the variable by undoing the operations or using inverse operations.
Page 10 Problem 37 Answer
We are given 3/4x+2=42/3
We are required to solve the equation.
We will isolate the variable from the constants to determine the solution of the given equation.
Given equation is 3/4x+2=42/3
We can write it as: 3/4x+2=14/3
Isolating the variable using inverse operations, we get
Subtracting 2 on both sides, we get
3/4x=8/3
Multiplying both sides by 4/3, we get
x=32/9
=35/9
The solution of the equation 3/4x+2=42/3 is x=35/9.
The steps can be shown as:
3/4x+2=14/3
3/4x=8/3
x=35/9
The strategy to solve any two-step equation is to isolate the variable by undoing the operations or using inverse operations.
Page 10 Problem 38 Answer
We are given −2/3b+2/5
=64/5
We are required to solve the equation.
We will isolate the variable from the constants to determine the solution of the given equation.
Given equation is −2/3b+2/5=64/5
Rewriting the equation, we get
−2/3b+2/5=34/5
Isolating the variable using inverse operations, we get
Subtracting 2/5 on both sides, we get
−2/3b=32/5
Multiplying both sides by −3/2, we get
b=−96/10
The solution of equation −2/3b+2/5=64/5 is
b=−96/10
The step can be shown as
−2/3b+2/5=64/5
−2/3b=32/5
b=−96/10
The strategy to solve any two-step equation is to isolate the variable by undoing the operations or using inverse operations.
Page 10 Problem 39 Answer
We are given −t/5−9=21
We are required to solve the equation.
We will isolate the variable from the constants to determine the solution of the given equation.
Given equation is −t/5−9=21
Isolating the variable using inverse operations, we get
Adding 9 on both sides, we get
−t/5=30
Multiplying both sides by −5, we get
t=−150
The solution of the equation −t/5−9=21 is t=−150
The steps can be shown as
−t/5
−9=21
−t/5=30
t=−150
The strategy to solve any two-step equation is to isolate the variable by undoing the operations or using inverse operations.
Page 10 Problem 40 Answer
We are given 2=2.27−s/4
We are required to solve the equation.
We will isolate the variable from the constants to determine the solution of the given equation.
Give equation is 2=2.27−s/4
To isolate the variable using inverse operations, we get
Subtracting 2.27 from both sides, we get −0.27=−s/4
Multiplying both sides by −4, we get 1.08=s
The solution of the equation 2=2.27−s/4 is s=1.08
The steps can be shown as
2=2.27−s/4
−0.27=−s/4
1.08=s
The strategy to solve any two-step equation is to isolate the variable by undoing the operations or using inverse operations.
Page 10 Problem 41 Answer
To solve: The given two step equation 12m−7=139
The given two step equation 12m−7=139
Adding 7 both sides
⇒12m−7+7=139+7
⇒12m=146
Dividing both sides by 12
⇒12m/12
=146/12
⇒m=12.16
General strategies to solve the equation are the inverse operation method and elimination method.
The value of m is obtained from the equation by using addition method of equality and then using division property of equality.
Page 10 Problem 42 Answer
To solve: The given two step equation 121.1=−19.3−4d
The given equation121.1=−19.3−4d
Adding 19.3 both sides
⇒121.1+19.3=−19.3+19.3−4d
⇒140.4=−4d
Dividing both sides by −4
⇒140.4/−4
=−4d/−4
⇒d=−35.1
General strategies to solve the equation are the inverse operation method and elimination method.
The value of d is obtained from the equation by using addition method of equality and then using division property of equality.
Page 10 Problem 43 Answer
To solve: The given two step equation−23z+234=970
The given two step equation −23z+234=970
Subtracting 234 from both sides
⇒−23z+234−234=970−234
⇒−23z=736
Dividing both sides by −23
⇒−23z/−23
=736/−23
⇒z=−32
General strategies to solve the equation are the inverse operation method and elimination method.
The value of z is obtained from the equation by using subtraction method of equality and then using division property of equality.
Page 10 Problem 44 Answer
To solve: The given two step equation7865=345−5d
⇒7865−345-5d
Subtracting 345 from both sides
⇒7865−345=345−345−5d
⇒7520=−5d
Dividing both sides by −5
⇒7520/−5
=−5d/−5
⇒d=−1504
General strategies to solve the equation are inverse operation method and elimination method .
The value of d is obtained from the equation by using subtraction method of equality and then using division property of equality.
Page 12 Problem 45 Answer
Given: An equation 2x−6=22
To find: The solution to the given equation.
Given : 2x−6=22
Adding 6 to both sides
⇒2x−6+6=22+6
⇒2x=28
Dividing by 2 both sides
⇒2x/2
=28/2
⇒x=14
The final solution to the equation is obtained by using addition property of equality in the first step and then using division property of equality in the second step.