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3 years ago

Aaron, Ben and Caleb all have grabbed some lollies from a jar. Ben has twice as

Mathematics

2 answers:

RoseWind [281]3 years ago

8 0

Answer:

Step-by-step explanation:

So, to start off with, find all of the variables in the equation.

Aaron: x

Ben: 2x

C: (5 + 2x)

65 is how many lollies we they have altogether.

So now we start to make the equation.

x + 2x + 5 + 2x = 65.

Add like terms.

5x + 5 = 65.

65 - 5 = 60

5x = 60

60/5 = 12

Aaron = 12 lollies

Now find the amount of bens lollies

2(12) = 24

Ben = 24 lollies

24 + 5 = 29

Caleb = 29 lollies.

Thepotemich [5.8K]3 years ago

4 0

Answer:

Aaron=12 Ben=24 Caleb=29

Step-by-step explanation:

Aaron=x

Ben=2x

Caleb=2x+5

x+2x+2x+5=65

5x+5=65

-5 on both sides

5x=60

divide by 5 on both sides

x=12

You then substitute this answer with ben and Caleb

ben= 2x (2*12) =24

caleb= 2x+5 (24+5)=29

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Dafna1 [17]

Answer:

The mid-point (p + q , q + p) of AB is the same distance from the x-axis and the y-axis

Step-by-step explanation:

* Lets explain how to solve the problem

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equal coordinates

- Ex: point (4 , 4) is the same distance from the x-axis and the y-axis

because the distance from the x-axis to the point is 4 (y-coordinate)

and the distance from the y-axis and the point is 4 (x-coordinate)

- If (x , y) is the mid-point of a segment its endpoints are

$(x_{1},y_{1})$ and $(x_{2},y_{2})$ , then

$x=\frac{x_{1}+x_{2}}{2}$ and $y=\frac{y_{1}+y_{2}}{2}$

* Lets solve the problem

∵ Point A has coordinates (p , q)

∵ Point B has coordinates (p + 2q , q + 2p)

- The mid-point of AB is (x , y)

∵ $x=\frac{p+p+2q}{2}$

∴ $x=\frac{2p+2q}{2}$

- Take 2 as a common factor from the terms of the numerator

∴ $x=\frac{2(p+q)}{2}$

- Divide up and down by 2

∴ x = p + q

∵ $y=\frac{q+q+2p}{2}$

∴ $y=\frac{2q+2p}{2}$

- Take 2 as a common factor from the terms of the numerator

∴ $y=\frac{2(q+p)}{2}$

- Divide up and down by 2

∴ y = q + p

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- p + q is the same with q + p

∵ The x-coordinate of the mid point of AB is p + q

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3 years ago

One winter morning the temperature was -2 C. By 11:00 am the temperature hd decreased by 3 degrees. At 4:00 pm reached 0 degrees

Temka [501]

Answer:

The temperature rose by 5 degrees celcius between 11 am and 4 pm

Step-by-step explanation:

Here in this question, we want to find the integer which represents the temperature change from 11 am to 4 pm.

Initially, in the morning, the temperature was -2 . By 11, it had decreased by 3 degrees. What this means is that the temperature went down further by 3 degrees.

Mathematically, the temperature at 11 will be -2-3 = -5 degrees celcius

Now at 4 pm, the temperature went from -5 to 0 degrees.

The temperature change between 11 am and 4 pm will be; 0-(-5) = 0 + 5 = 5 degrees celcius

This means that the temperature rose by 5 degrees celcius between 11 am and 4 pm

7 0

3 years ago

Y=x^2-12x+45 vertex form and coordinate vertex

zysi [14]

Best Answer

 x2-12x-45=0 Two solutions were found : x = 15 x = -3

Step by step solution :Step 1 :Skip Ad
Trying to factor by splitting the middle term

1.1 Factoring x2-12x-45

The first term is, x2 its coefficient is 1 .
The middle term is, -12x its coefficient is -12 .
The last term, "the constant", is -45 

Step-1 : Multiply the coefficient of the first term by the constant 1 • -45 = -45

Step-2 : Find two factors of -45 whose sum equals the coefficient of the middle term, which is -12 .

-45 + 1 = -44 -15 + 3 = -12 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -15 and 3
 x2 - 15x + 3x - 45

Step-4 : Add up the first 2 terms, pulling out like factors :
 x • (x-15)
 Add up the last 2 terms, pulling out common factors :
 3 • (x-15)
Step-5 : Add up the four terms of step 4 :
 (x+3) • (x-15)
 Which is the desired factorization

Equation at the end of step 1 : (x + 3) • (x - 15) = 0 Step 2 :Theory - Roots of a product :

2.1 A product of several terms equals zero.

When a product of two or more terms equals zero, then at least one of the terms must be zero.

We shall now solve each term = 0 separately

In other words, we are going to solve as many equations as there are terms in the product

Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

2.2 Solve : x+3 = 0

Subtract 3 from both sides of the equation :
 x = -3

Solving a Single Variable Equation :

2.3 Solve : x-15 = 0

Add 15 to both sides of the equation :
 x = 15

Supplement : Solving Quadratic Equation DirectlySolving x2-12x-45 = 0 directly

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Parabola, Finding the Vertex :

3.1 Find the Vertex of y = x2-12x-45

Parabolas have a highest or a lowest point called the Vertex . Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) . We know this even before plotting "y" because the coefficient of the first term, 1 , is positive (greater than zero).

Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.

Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.

For any parabola,Ax2+Bx+C,the x -coordinate of the vertex is given by -B/(2A) . In our case the x coordinate is 6.0000 

Plugging into the parabola formula 6.0000 for x we can calculate the y -coordinate :
 y = 1.0 * 6.00 * 6.00 - 12.0 * 6.00 - 45.0
or y = -81.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for : y = x2-12x-45
Axis of Symmetry (dashed) {x}={ 6.00}
Vertex at {x,y} = { 6.00,-81.00}
x -Intercepts (Roots) :
Root 1 at {x,y} = {-3.00, 0.00}
Root 2 at {x,y} = {15.00, 0.00}

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3 years ago

If T: (x, y) → (x + 6, y + 4), then T-1: (x,y) → _____.

alexgriva [62]

T is a linear transformation from R²→R² with basis {(1,0),(0,1)}
T: (x,y)→(x+6,y+4)

A function from one vector space to another that preserves the underlying (linear) structure of each vector space is called a linear transformation.

Then the vector (1,0) goes to (1+6,4)=(7,4)=7(1,0)+4(0,1)

and the vector (0,1) goes to (6,1+4)=(6,5)=6(1,0)+5(0,1)

So, the matrix of the transformation is

$\left[\begin{array}{ccc}7&6\\4&5\end{array}\right]$

The inverse of the matrix is

$\left[\begin{array}{ccc}\frac{5}{11}&\frac{-6}{11}\\ \\\frac{-4}{11}&\frac{7}{11}\end{array}\right]$

So, the Inverse Transformation is given by

$T^{-1}(x,y)=\left[\begin{array}{ccc}\frac{5}{11}&\frac{-6}{11}\\ \\\frac{-4}{11}&\frac{7}{11}\end{array}\right]\left[\begin{array}{ccc}x\\y\end{array}\right] =(\frac{5x-6y}{11}, \frac{-4x+7y}{11})$