Solve the system of equations
3x-y=11
-5x+y=12
2.
Perform the indicated operation and simplify (5/(x-2)+(4(x^2-2x)
3.simpify and write with only positive exponents (7x^3y^-2)/(x^-4y^9)^-3
4.
find the solution of the inequality
2x-11 is less than or equal to -4(5x-3)
5. the length of a recangle is 5 feet less than twice the width. the area is 25 square feet. using w as the variable. write an equation that can be used to calculate the width.
(2w^2-5)
Any help would be appreciated. This is for my study guide in order to do well on my test. Someone please help.
#1
From the first, y=3x-11
Plug that into the second:
-5x+3x-11 = 12
-2x = 23
x = -23/2
y = -91/2
Or, you can just add the equations to eliminate the y terms:
-5x = 23
...
#2 Your parentheses are unbalanced. If you mean
5/(x-2) + 4/(x^2-2x)
5/(x-2) + 4/(x(x-2))
5x/(x(x-2)) + 4/(x(x-2))
(5x+4)/(x(x-2))
(5x+4)/(x^2-2x)
#3
(7x^3y^-2)/(x^-4y^9)^-3
(x^-4y^9)^-3 = x^12 y^-27
Since you're dividing by that, you have
(7x^3y^-2)*(x^-12 y^27)
7x^-9 y^25
7y^25/x^9
#4.
2x-11 <= -4(5x-3)
2x-11 <= -20x + 12
22x <= 23
x <= 23/22
#5.
length is 2w-5
area is 25, so
w(2w-5) = 25
2w^2-5w = 25
2w^2 - 5w - 25 = 0
(2w+5)(w-5) = 0
w=5 (-5/2 is not a usable width)
the rectangle is a square 5x5
1. To solve the system of equations:
3x - y = 11
-5x + y = 12
We can use the method of substitution to eliminate one of the variables. First, let's solve one equation for one variable in terms of the other.
From the second equation, we can isolate y:
y = 5x + 12
Substitute this value of y into the first equation:
3x - (5x + 12) = 11
3x - 5x - 12 = 11
-2x - 12 = 11
-2x = 11 + 12
-2x = 23
x = 23 / -2
x = -11.5
Now, we can substitute x = -11.5 into either of the original equations to find the value of y. Taking the first equation:
3(-11.5) - y = 11
-34.5 - y = 11
y = -34.5 - 11
y = -45.5
Therefore, the solution to the system of equations is x = -11.5 and y = -45.5.
2. To perform the indicated operation and simplify:
(5/(x-2)) + (4(x^2-2x))
First, let's simplify the expression within the parentheses:
4(x^2 - 2x) = 4x^2 - 8x
Now, we can rewrite the expression with the common denominator:
(5/(x-2)) + (4x^2 - 8x)/(x-2))
To add the fractions, we need a common denominator, which is (x - 2).
Combine the numerators:
(5 + 4x^2 - 8x) / (x - 2)
Simplify the numerator:
(4x^2 - 8x + 5) / (x - 2)
So, the simplified expression is (4x^2 - 8x + 5) / (x - 2).
3. To simplify and write with only positive exponents:
(7x^3y^-2) / (x^-4y^9)^-3
First, let's simplify the expression inside the parentheses:
(x^-4y^9)^-3 = x^12 * y^-27
Now, let's substitute this back into the original expression:
(7x^3y^-2) / (x^12 * y^-27)
When we divide two expressions with the same base, we subtract their exponents:
7x^(3 - 12) * y^(-2 + 27)
Simplify the exponents and combine the terms:
7x^-9 * y^25 / (1 * y^25)
Since any number divided by 1 is itself, we can remove the denominator:
7x^-9 * y^25 * y^-25
When we multiply two expressions with the same base, we add their exponents:
7x^-9 * y^(25 - 25)
Simplify further:
7 * x^-9 * y^0
Since any number raised to the 0th power is 1, we can remove y:
7 * x^-9
Finally, we need to convert the negative exponent to a positive exponent:
7 / x^9
So, the simplified expression with only positive exponents is 7 / x^9.
4. To find the solution of the inequality:
2x - 11 ≤ -4(5x - 3)
First, distribute the -4 on the right side:
2x - 11 ≤ -20x + 12
Next, simplify by combining like terms:
2x + 20x ≤ 12 + 11
22x ≤ 23
Divide both sides of the inequality by 22 to isolate x:
x ≤ 23 / 22
Therefore, the solution to the inequality is x ≤ 23/22.
5. To write an equation that can be used to calculate the width of a rectangle:
The length of a rectangle is 5 feet less than twice the width. Let's denote the width as w.
Twice the width is 2w, and 5 feet less than that is 2w - 5. We know that the area of a rectangle is given by length times width.
Therefore, the equation for the area is:
Length * Width = Area
(w)(2w - 5) = 25
So, the equation that can be used to calculate the width is 2w^2 - 5w = 25.