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?
Let's say the width of the rectangle is "W" meters.
The length of the rectangle should be 2W - 4, according to the problem.
The area of the rectangle is length times width, which we can write as: W * (2W - 4) = 96.
Expanding the equation: 2W^2 - 4W = 96.
Subtracting 96 from both sides, we get: 2W^2 - 4W - 96 = 0.
Dividing the equation by 2, we get: W^2 - 2W - 48 = 0.
Factoring the quadratic equation, we have: (W - 8)(W + 6) = 0.
So, W - 8 = 0 or W + 6 = 0.
If W - 8 = 0, then the width of the rectangle is W = 8 meters.
If W + 6 = 0, then the width of the rectangle is W = -6, which is not a valid width.
Since the width cannot be negative, we can conclude that the width of the rectangle is 8 meters.
The length of the rectangle is 2W - 4 = 2 * 8 - 4 = 12 meters.
Therefore, the length of the rectangle is 12 meters and the width is 8 meters.
An expression for the length of the rectangle in terms of the width would be ______
An expression for the length of the rectangle in terms of the width would be: length = 2 * width - 4.
3. Match the equation with its solution(s).(5 points)
3i+2i−7=18
3i+5=2i−7
3(2i−7)=15
3(2i+7)=6i+21
3i+5=3i+7
Solve the equation justifying each step with the correct reasoning.
2(x+8)=2x+8
(5 points)
To solve the equation 2(x + 8) = 2x + 8, we will apply the distributive property and then simplify step by step:
1. Start with the given equation: 2(x + 8) = 2x + 8
2. Apply the distributive property: 2 * x + 2 * 8 = 2x + 8
3. Simplify by multiplying: 2x + 16 = 2x + 8
4. Notice that we have 2x on both sides of the equation.
Subtraction property of equality states that if a = b, then a - b = 0.
So, we can subtract 2x from both sides of the equation.
5. Subtract 2x from both sides: 2x - 2x + 16 = 2x - 2x + 8
Simplify: 0 + 16 = 0 + 8
We are left with: 16 = 8
6. Since 16 is not equal to 8, the equation is inconsistent.
Therefore, there is no solution for the given equation.
In summary, the equation 2(x + 8) = 2x + 8 has no solution.
5. Match the description of the one variable equation with the number of solutions it will have.(4 points)
x−7=7−x
3(x+5)=3x+5
10−x=25
2(x+4)=2x+5+3
Matching the description of the one-variable equation with the number of solutions it will have:
1. x - 7 = 7 - x
This equation has one solution. It simplifies to: 2x = 14, and then x = 7.
2. 3(x + 5) = 3x + 5
This equation has infinitely many solutions. When simplified, all terms involving x cancel out, resulting in a true statement of 0 = 0.
3. 10 - x = 25
This equation has no solution. When simplified, it leads to a contradiction: -x = 15, which is not possible.
4. 2(x + 4) = 2x + 5 + 3
This equation has no solutions. When simplified, it leads to a contradiction: 2x + 8 = 2x + 8, which is always true.
In summary:
1. One solution
2. Infinitely many solutions
3. No solution
4. No solution
6. 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?
24n+2≥37
37>2n+24
37<2n+24
37≥2n+24
To describe the scenario where a student wants to purchase a calculator costing $24 and some notebooks at $2 each while having a budget of $37, we can write the inequality:
Total cost of calculator + Total cost of notebooks ≤ Budget
The variable n represents the number of notebooks the student buys.
The expression for the total cost of notebooks is 2n, as each notebook costs $2.
So, the inequality that describes this scenario is:
24 + 2n ≤ 37
Therefore, the correct inequality is: 24 + 2n ≤ 37.
Solve for b in the following equation: A=1/2(a+b)
To solve for b in the equation A = 1/2(a + b), we can follow these steps:
1. Start with the given equation: A = 1/2(a + b).
2. Multiply both sides of the equation by 2 to eliminate the fraction: 2A = (a + b).
3. Distribute on the right side: 2A = a + b.
4. Rearrange the equation by subtracting 'a' from both sides: 2A - a = b.
5. The final equation is: b = 2A - a.
Therefore, b = 2A - a.
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)
To determine how many minutes it will take Grandma to have only 30 weeds remaining in the garden, we can use the following steps:
1. Start with the given information:
- Initial number of weeds = 250
- Rate of weed removal = 5 weeds per minute
- Target number of weeds = 30
2. Determine the difference between the initial number of weeds and the target number of weeds:
Difference = Initial number of weeds - Target number of weeds
Difference = 250 - 30
Difference = 220
3. Divide the difference by the rate of weed removal to find the number of minutes required:
Number of minutes = Difference / Rate of weed removal
Number of minutes = 220 / 5
Number of minutes = 44
Therefore, it will take Grandma approximately 44 minutes to have only 30 weeds remaining in the garden.
Part A: Solve the equation for x showing all steps for credit: ax−5=18
To solve the equation ax - 5 = 18 for x, we will follow the steps below:
1. Start with the given equation: ax - 5 = 18.
2. Add 5 to both sides of the equation to isolate the term with x:
ax - 5 + 5 = 18 + 5.
Simplifying, we have: ax = 23.
3. Divide both sides of the equation by a to solve for x:
\(\frac{ax}{a}\) = \(\frac{23}{a}\).
Simplifying, we get: x = \(\frac{23}{a}\).
Therefore, the solution to the equation ax - 5 = 18 is x = \(\frac{23}{a}\).
Matching the equations with their solutions:
1. 3i + 2i - 7 = 18
Solution: No solution. The equation is inconsistent.
2. 3i + 5 = 2i - 7
Solution: i = -12
3. 3(2i - 7) = 15
Solution: i = 4
4. 3(2i + 7) = 6i + 21
Solution: i = 2
5. 3i + 5 = 3i + 7
Solution: No solution. The equation is inconsistent.