1.The square of x is equal to 4 times the square od y. If x is 1 more than twice y, what is the value of x?
A) -4
B) -.5
C) -.25
D) .25
E) .5
2.In the xy-coordinate plane, lines L an Q are perpendicular. If line L contains the points (0,0) and (2,1), and line Q contains the points ( 2,1) AND (0,T) What is the value of T?
A)-3
B)-2
C)2
D)3
E)5
3. For all positivw integers x, let xÄ
be defined to be (x-1)(x+1). Which of the following is equal to 6Ä-5Ä?
A)2Ä= 1Ä
B)3Ä= 2Ä
C)4Ä= 3Ä
D)5Ä= 4Ä
E)6Ä= 5Ä
Do you have any thinking on these? Surely you do not want us to give you answers.
1. X^2= 4Y^2 and x=2y+1
Sooo i substitute
(2y+1)^2= 4y^2
Find the square root of both sides
2y+1= 4y
Solve for y
2y+1=4y
1=4y
.25=y
Substitue into X=2y=1
x=2(.25)+1
x=.5+1
x=1.5
But that's not an option.What did i do wrong?
1. To solve this problem, we can set up two equations using the information given.
First, we have the equation: x^2 = 4y^2.
Second, we have the equation: x = 2y + 1.
Substituting the second equation into the first equation, we get: (2y + 1)^2 = 4y^2.
Expanding and simplifying, we get: 4y^2 + 4y + 1 = 4y^2.
Canceling out the 4y^2 terms, we get: 4y + 1 = 0.
Solving for y, we get y = -1/4.
Substituting this value of y back into the second equation, we can find x: x = 2(-1/4) + 1 = -1/2 + 1 = 1/2.
Therefore, the value of x is 1/2, which corresponds to option E) 0.5.
2. We are given that lines L and Q are perpendicular, which means their slopes are negative reciprocals of each other. Since line L passes through the points (0,0) and (2,1), we can find the slope of L using the formula: slope = Δy/Δx = (1-0)/(2-0) = 1/2.
Since line Q contains the points (2,1) and (0,T), we can find its slope using the same formula: slope = Δy/Δx = (1-T)/(2-0).
Since the two lines are perpendicular, their slopes must be negative reciprocals. Therefore, we have the equation: 1/2 = -(2-0)/(1-T).
Simplifying this equation, we get: 1/2 = 2/(1-T).
Cross-multiplying, we get: 1(1-T) = 2(2).
Simplifying further, we get: 1 - T = 4.
Solving for T, we get T = -3.
Therefore, the value of T is -3, which corresponds to option A) -3.
3. To find the value of 6Ä - 5Ä, we need to substitute the definition of Ä into the expression.
The definition of Ä is (x-1)(x+1).
So, we can rewrite 6Ä - 5Ä as 6(x-1)(x+1) - 5(x-1)(x+1).
Expanding this expression, we get: (6x - 6)(x + 1) - (5x - 5)(x + 1).
Further simplifying, we get: 6x^2 + 6x - 6 - 5x^2 - 5x - 5x + 5.
Combining like terms, we get: x^2 + x + 1.
Therefore, 6Ä - 5Ä is equal to x^2 + x + 1.
None of the given options match x^2 + x + 1, so there is no appropriate answer.