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.