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3 years ago

Given: 3x = 5 (x − 4) Prove: x = 10

Mathematics

1 answer:

damaskus [11]3 years ago

4 0

Answer:

see the procedure

Step-by-step explanation:

we have

$3x=5(x-4)$

step 1

$3x=5(x-4)$ ------> given problem

step 2

Applying the Distributive Property right side

$3x=5(x)-5(4)$

$3x=5x-20$

step 3

Applying the Subtraction Property of Equality

Subtract (3x-20) both sides

$3x-(3x-20)=5x-20-(3x-20)$

$20=2x$

step 4

Applying Division Property of Equality

Divide by 2 both sides

$20\2=x$

Rewrite

$x=10$

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Answer:

slope = 2

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3 years ago

A study is given in which scientists examined data on mean sea surface temperatures (in degrees Celsius) and mean coral growth (

Natalka [10]

Answer:

a) $\bar x= \frac{\sum x_i}{n}=\frac{211.81}{7}=30.26$

$\bar y= \frac{\sum y_i}{n}=\frac{17.63}{7}=2.52$

$r=\frac{7(533.1304)-(211.81)(17.63)}{\sqrt{[7(6410.063) -(211.81)^2][7(44.5511) -(17.63)^2]}}=-0.85$

$b = \frac{S_{xy}}{S_{xx}}$

Where:

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=6410.063-\frac{211.81^2}{7}=0.9950$

$S_{yy}=\sum_{i=1}^n y^2_i -\frac{(\sum_{i=1}^n y_i)^2}{n}=44.5511-\frac{17.63^2}{7}=0.1487$

$b = \frac{S_{xy}}{S_{xx}}$

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i){n}}=533.1304-\frac{211.81*17.63}{7}=-0.3282$

And the slope would be:

$m=\frac{-0.3282}{0.9950}=-0.330$

And we can find the intercept using this:

$b=\bar y -m \bar x=2.52-(-0.3299*30.26)=12.502$

So then the linear model would be:

$y =-0.330 x + 12.502$

b) > x<-c(29.67, 29.87, 30.15, 30.21, 30.47, 30.64, 30.80)

> y<-c(2.64,2.59, 2.69, 2.60, 2.48, 2.38, 2.25)

> lm<- lm(y~x)

> lm

Call:

lm(formula = y ~ x)

Coefficients:

(Intercept) x

12.5009 -0.3299

And the model is given by:

$y= -0.330 x +12.501$

c) Every increase of one degree Celsius means about -0.330 fewer mean millimeters of coral growth per year.

Step-by-step explanation:

For this case we have the following data:

X: 29.67 29.87 30.15 30.21 30.47 30.64 30.80

Y: 2.64 2.59 2.69 2.60 2.48 2.38 2.25

Part a

$\bar x= \frac{\sum x_i}{n}=\frac{211.81}{7}=30.26$

$\bar y= \frac{\sum y_i}{n}=\frac{17.63}{7}=2.52$

The correlation coefficient is a "statistical measure that calculates the strength of the relationship between the relative movements of two variables". It's denoted by r and its always between -1 and 1.

And in order to calculate the correlation coefficient we can use this formula:

$r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^2 -(\sum x)^2][n\sum y^2 -(\sum y)^2]}}$

For our case we have this:

n=7 $\sum x = 211.81, \sum y = 17.63, \sum xy = 533.1304, \sum x^2 =6410.063, \sum y^2 =44.5511$

$r=\frac{7(533.1304)-(211.81)(17.63)}{\sqrt{[7(6410.063) -(211.81)^2][7(44.5511) -(17.63)^2]}}=-0.85$

We can calculate the slope for the regression model with this formula:

$b = \frac{S_{xy}}{S_{xx}}$

Where:

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=6410.063-\frac{211.81^2}{7}=0.9950$

$S_{yy}=\sum_{i=1}^n y^2_i -\frac{(\sum_{i=1}^n y_i)^2}{n}=44.5511-\frac{17.63^2}{7}=0.1487$

And if we replace we got:

$b = \frac{S_{xy}}{S_{xx}}$

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i){n}}=533.1304-\frac{211.81*17.63}{7}=-0.3282$

And the slope would be:

$m=\frac{-0.3282}{0.9950}=-0.330$

And we can find the intercept using this:

$b=\bar y -m \bar x=2.52-(-0.3299*30.26)=12.502$

So then the linear model would be:

$y =-0.330 x + 12.502$

Part b

For this case we use the following R code:

> x<-c(29.67, 29.87, 30.15, 30.21, 30.47, 30.64, 30.80)

> y<-c(2.64,2.59, 2.69, 2.60, 2.48, 2.38, 2.25)

> lm<- lm(y~x)

> lm

Call:

lm(formula = y ~ x)

Coefficients:

(Intercept) x

12.5009 -0.3299

And the model is given by:

$y= -0.330 x +12.501$

Part c

For this case the best interpretation is:

Every increase of one degree Celsius means about -0.330 fewer mean millimeters of coral growth per year.

8 0

3 years ago