1. What is the distance between the points (1, 4) and (4, 8)?
I said Distance = 5
2. Find the slope between (1, 4) and (4, 8)
I said 4/3
3. Are the expressions (4x+4)/4 and x+1 equivalent?
I said Yes
4. Simplify (x+y)2
x2 + 2xy + y2
5. Is (x – y )2 = x2 – y2?
No. It is equal to x2 - 2xy + y2
6. Identify each equation as true or false. If it’s true, explain why. If it’s false, provide a counterexample.
a. (4y+4)/4=y
False it is equal to y + 1
b.(3x+3y)/3=x+y
True
all look ok to me.
1. To find the distance between two points in the coordinate plane, you can use the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
For the given points (1, 4) and (4, 8), plug in the values into the formula:
Distance = √((4 - 1)^2 + (8 - 4)^2)
= √(3^2 + 4^2)
= √(9 + 16)
= √25
= 5
So, the distance between the points (1, 4) and (4, 8) is indeed 5.
2. The slope between two points in the coordinate plane can be found using the slope formula:
Slope = (y2 - y1) / (x2 - x1)
Using the points (1, 4) and (4, 8), plug in the values into the formula:
Slope = (8 - 4) / (4 - 1)
= 4 / 3
So, the slope between the points (1, 4) and (4, 8) is indeed 4/3.
3. To determine if two expressions are equivalent, you need to simplify and compare them.
The expressions given are (4x+4)/4 and x+1. Simplifying them:
(4x+4)/4 = 4(x+1)/4 = x+1
The simplified form of both expressions is x+1. Therefore, they are equivalent.
4. To simplify the expression (x+y)^2, you need to apply the concept of expanding a binomial squared.
(x+y)^2 = (x+y)(x+y) = x(x+y) + y(x+y)
= x^2+xy + xy+y^2
= x^2 + 2xy + y^2
Hence, the simplified form of (x+y)^2 is x^2 + 2xy + y^2.
5. The equation (x-y)^2 is not equal to x^2 - y^2.
To expand (x-y)^2, you need to apply the concept of expanding a binomial squared:
(x-y)^2 = (x-y)(x-y) = x(x-y) - y(x-y)
= x^2 - xy - xy + y^2
= x^2 - 2xy + y^2
Therefore, (x-y)^2 simplifies to x^2 - 2xy + y^2, not x^2 - y^2.
6. a. The equation (4y+4)/4 = y is false.
To show it, we can simplify the left side of the equation:
(4y+4)/4 = y
Dividing both sides by 4, we get:
4y + 4 = 4y
But this cannot be true because 4y + 4 will always be greater than 4y. Therefore, the equation is false.
b. The equation (3x+3y)/3 = x+y is true.
Similarly, we can simplify the left side of the equation:
(3x+3y)/3 = x+y
Dividing both sides by 3, we get:
3x + 3y = 3x + 3y
This equation is true because the left side is equal to the right side. Therefore, the equation is true.