The length of a rectangle is four meters less than twice its width.
If the area of the rectangle is 96 m^2, what is the length and the width?
(3 points)
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An expression for the length of the rectangle in terms of the width would be Response area
The formula for the area of a rectangle is Response area
Using trial and error, if the area is 96 m^2, then the length and width are Response area
An expression for the length of the rectangle in terms of the width would be Response: 2w - 4.
The formula for the area of a rectangle is Response: length * width.
Using trial and error, if the area is 96 m^2, then the length and width are Response: length = 12 m and width = 8 m.
The length of a rectangle is four meters less than twice its width.
If the area of the rectangle is 96 m^2, what is the length and the width?
An expression for the length of the rectangle in terms of the width would be
The expression for the length of the rectangle in terms of the width would be: length = 2w - 4.
The formula for the area of a rectangle is
The formula for the area of a rectangle is: area = length * width.
Using trial and error, if the area is 96 m^2, then the length and width are
Using trial and error, if the area is 96 m^2, we can find possible values for the length and width that satisfy the given conditions.
First, let's express the length in terms of the width:
length = 2w - 4
Now, we can set up a trial and error table to find the values of length and width that give an area of 96 m^2:
Width (w) | Length (2w - 4) | Area (length * width)
----------------------------------------------------
1 | 2(1) - 4 = -2 | -2
2 | 2(2) - 4 = 0 | 0
3 | 2(3) - 4 = 2 | 6
4 | 2(4) - 4 = 4 | 16
5 | 2(5) - 4 = 6 | 30
6 | 2(6) - 4 = 8 | 48
7 | 2(7) - 4 = 10 | 70
8 | 2(8) - 4 = 12 | 96
9 | 2(9) - 4 = 14 | 126
From the table, we can see that when the width is 8 meters, the length is 2(8) - 4 = 12 meters. Additionally, the area of the rectangle is 96 m^2.
Therefore, the length of the rectangle is 12 meters and the width is 8 meters.
3(2i−7)=15
To solve the equation 3(2i - 7) = 15, we can proceed as follows:
First, distribute the 3 to the terms inside the parentheses:
6i - 21 = 15
Next, isolate the variable term by adding 21 to both sides of the equation:
6i = 15 + 21
Simplifying the right side gives:
6i = 36
Finally, divide both sides of the equation by 6 to solve for i:
i = 36/6
Simplifying the right side gives:
i = 6
Therefore, the solution to the equation 3(2i - 7) = 15 is i = 6.
3i+5=2i−7
To solve the equation 3i + 5 = 2i - 7, we can proceed as follows:
First, subtract 2i from both sides of the equation to isolate the variable:
3i - 2i + 5 = 2i - 2i - 7
Simplifying gives:
i + 5 = -7
Next, subtract 5 from both sides of the equation:
i + 5 - 5 = -7 - 5
Simplifying gives:
i = -12
Therefore, the solution to the equation 3i + 5 = 2i - 7 is i = -12.
3i+2i−7=18
To solve the equation 3i + 2i - 7 = 18, we can combine like terms first:
(3i + 2i) - 7 = 18
5i - 7 = 18
Next, isolate the variable term by adding 7 to both sides of the equation:
5i - 7 + 7 = 18 + 7
Simplifying gives:
5i = 25
Finally, divide both sides of the equation by 5 to solve for i:
i = 25/5
Simplifying further gives:
i = 5
Therefore, the solution to the equation 3i + 2i - 7 = 18 is i = 5.
3i+5=3i+7
To solve the equation 3i + 5 = 3i + 7, we can see that the variable "i" cancels out on both sides of the equation. This results in 5 = 7, which is not a true statement.
Therefore, there is no solution to the equation 3i + 5 = 3i + 7.
3(2i+7)=6i+21
To solve the equation 3(2i+7) = 6i+21, we can start by distributing the 3 to the terms inside the parentheses:
6i + 21 = 6i + 21
Notice that the variable "i" cancels out on both sides of the equation. This results in 21 = 21, which is a true statement.
Therefore, the equation 3(2i+7) = 6i+21 is true for any value of "i". In this case, the equation does not give a unique solution for "i".
so infinite solutions?
Yes, that's correct. The equation 3(2i+7) = 6i+21 has infinitely many solutions. This means that any value of "i" will satisfy the equation, resulting in an infinite number of possible solutions.
Solve the equation justifying each step with the correct reasoning.
2(x+8)=2x+8
Step 1: [ ] property to get [ ] simplified equation
Step 2: [ ] Property to get [ ] simplified equation
For this equation, there is/are [ ]
Step 1: Apply the distributive property to the left side of the equation.
2(x + 8) = 2x + 8
This simplifies to:
2x + 16 = 2x + 8
Step 2: Notice that the variable "x" appears on both sides of the equation. By subtracting 2x from both sides, we can eliminate the variable on the right side.
2x + 16 - 2x = 2x + 8 - 2x
Simplifying this gives:
16 = 8
However, 16 is not equal to 8. Thus, we have arrived at a contradiction.
Therefore, for this equation, there are no solutions.
x−7=7−x
To solve the equation x - 7 = 7 - x, we can follow these steps:
Step 1: Add x to both sides of the equation to eliminate the variable on the right side:
x - 7 + x = 7 - x + x
Simplifying gives:
2x - 7 = 7
Step 2: Add 7 to both sides of the equation to isolate the variable term:
2x - 7 + 7 = 7 + 7
Simplifying gives:
2x = 14
Step 3: Divide both sides of the equation by 2 to solve for x:
(2x) / 2 = 14 / 2
Simplifying gives:
x = 7
Therefore, the solution to the equation x - 7 = 7 - x is x = 7.
3(x+5)=3x+5
To solve the equation 3(x + 5) = 3x + 5, we can follow these steps:
Step 1: Apply the distributive property to the left side of the equation:
3 * x + 3 * 5 = 3x + 5
This simplifies to:
3x + 15 = 3x + 5
Step 2: Notice that the variable "x" appears on both sides of the equation. By subtracting 3x from both sides, we can eliminate the variable on the right side:
3x + 15 - 3x = 3x + 5 - 3x
Simplifying this gives:
15 = 5
However, 15 is not equal to 5. Thus, we have arrived at a contradiction.
Therefore, for this equation, there are no solutions.
10−x=25
To solve the equation 10 - x = 25, we can follow these steps:
Step 1: Subtract 10 from both sides of the equation to isolate the variable term:
10 - x - 10 = 25 - 10
Simplifying gives:
-x = 15
Step 2: Multiply both sides of the equation by -1 to solve for x:
(-1) * (-x) = (-1) * 15
Simplifying gives:
x = -15
Therefore, the solution to the equation 10 - x = 25 is x = -15.
2(x+4)=2x+5+3
To solve the equation 2(x + 4) = 2x + 5 + 3, we can follow these steps:
Step 1: Apply the distributive property to the left side of the equation:
2 * x + 2 * 4 = 2x + 5 + 3
This simplifies to:
2x + 8 = 2x + 8
Step 2: Notice that the variable "x" appears on both sides of the equation. By subtracting 2x from both sides, we can eliminate the variable on the right side:
2x + 8 - 2x = 2x + 8 - 2x
Simplifying this gives:
8 = 8
Since 8 is equal to 8, the equation is true.
Step 3: Therefore, the solution to the equation 2(x + 4) = 2x + 5 + 3 is that it is true for any value of x. In this case, the equation does not give a unique solution for x.
A student wants to purchase some new school supplies. He wants to buy a calculator that costs $24 and some notebooks for school. Each notebook costs $2. The student only has $37 to spend.
Let n represent the number of notebooks that he buys.
Which inequality describes this scenario?
The inequality that describes this scenario is:
24 + 2n ≤ 37
In this inequality, 24 represents the cost of the calculator, 2n represents the cost of the notebooks (since each notebook costs $2 and the student is buying n notebooks), and 37 represents the maximum amount of money the student has to spend.
are you sure
Apologies for the confusion. Let's correct the inequality.
Since the student wants to buy a calculator that costs $24 and n notebooks that cost $2 each, the total cost of the notebooks would be 2n.
The maximum amount the student has to spend is $37.
To represent this scenario using an inequality, we can write:
24 + 2n ≤ 37
In this inequality, 24 represents the cost of the calculator, 2n represents the cost of the notebooks, and 37 represents the maximum amount the student has to spend.
Therefore, the correct inequality for this scenario is 24 + 2n ≤ 37.
37≥2n+24
37<2n+24
37>2n+24
24n+2≥37
Apologies for the previous incorrect response.
The correct inequality that describes this scenario is:
37 ≥ 2n + 24
In this inequality, 37 represents the maximum amount the student has to spend, 2n represents the cost of the notebooks (each notebook costing $2 and the student buying n notebooks), and 24 represents the cost of the calculator.
Therefore, the student must ensure that the combined cost of the calculator and notebooks is less than or equal to $37, as represented by the inequality 37 ≥ 2n + 24.
Solve for b in the following equation: A=12(a+b)
b=2A+a
b=2A−a
b=12A−a
b=12A+a
To solve for b in the equation A = 12(a + b), we can follow these steps:
1. Distribute the 12 on the right side of the equation:
A = 12a + 12b
2. Move the 12a term to the left side by subtracting 12a from both sides:
A - 12a = 12a + 12b - 12a
Simplifying gives:
-A + 12a = 12b
3. Factor out 12 on the right side:
-A + 12a = 12(b)
4. Divide both sides of the equation by 12 to solve for b:
(-A + 12a) / 12 = 12b / 12
Simplifying gives:
(-A + 12a) / 12 = b
Therefore, the solution for b is:
b = (-A + 12a) / 12
Solve for b in the following equation: A=1/2(a+b)
b=2A+a
b=2A−a
b=1/2A−a
b=1/2A+a
To solve for b in the equation A = 1/2(a + b), we can follow these steps:
1. Multiply both sides of the equation by 2 to eliminate the fraction:
2 * A = 2 * (1/2(a + b))
Simplifying gives:
2A = a + b
2. Subtract a from both sides of the equation to isolate b:
2A - a = a + b - a
Simplifying gives:
2A - a = b
Therefore, the solution for b is:
b = 2A - a
−3x+1≤−47
To solve the inequality -3x + 1 ≤ -47, we can follow these steps:
1. Subtract 1 from both sides of the inequality:
-3x + 1 - 1 ≤ -47 - 1
Simplifying gives:
-3x ≤ -48
2. Divide both sides of the inequality by -3. Note that when dividing by a negative number, the inequality sign flips:
-3x / -3 ≥ -48 / -3
Simplifying gives:
x ≥ 16
Therefore, the solution to the inequality -3x + 1 ≤ -47 is x ≥ 16.
A student solves the following equation:
Problem: 2(x−3)+3x=19
Step 1: 2x−6+3x=19
Step 2: (2x+3x)−6=19
Step 3: 5x−6=19
Step 4: 5x−6+6=19+6
Step 5: 5x=25
Step 6: x=5
What property justifies going from step 3 to step 4?
(1 point)
Responses
Distributive Property
Addition Property of Equality
Substitution Property
Combine Like Terms
Commutative Property of Addition
The property that justifies going from step 3 to step 4 is the Addition Property of Equality.
In step 3, we have the equation 5x - 6 = 19. To isolate the variable term, we add 6 to both sides of the equation using the Addition Property of Equality.
This property allows us to perform the same operation (addition) to both sides of the equation, maintaining equality.
Therefore, the property used is the Addition Property of Equality.
A student solved the equation: x+2(x+1)=17
Step 1: x+2x+2=17
Step 2: 3x+2=17
Step 3: 3x=15
Step 4: x=45
Identify the property used to arrive at Step 1:
What step includes a mistake made from the previous step?
What should have been the correct answer for x ?
The property used to arrive at Step 1 is the Distributive Property.
The mistake is in Step 4. When solving Step 3, we should divide both sides of the equation by 3 to isolate the variable x.
Step 3: 3x = 15
Step 4: x = 15/3
Simplifying, we get:
x = 5
Therefore, the correct answer for x is 5.
Grandma is removing weeds from her garden. She estimates that there are 250 weeds in the garden at the moment, and she can remove weeds at a rate of 5 per minute. At this rate, how many minutes will it take her to only have 30 weeds remaining in the garden?(3 points)
Equation:
Solution:
Meaning of the solution in words:
Equation:
Let t represent the number of minutes it takes for Grandma to remove the weeds.
250 - 5t = 30
Solution:
To find the value of t, we need to solve the equation:
250 - 5t = 30
To do this, we can start by isolating the term
Solution:
Grandma is removing weeds from her garden. She estimates that there are 250 weeds in the garden at the moment, and she can remove weeds at a rate of 5 per minute. At this rate, how many minutes will it take her to only have 30 weeds remaining in the garden?(3 points)
Equation:
Solution:
Meaning of the solution in words: