Answer:
Part 38) [tex]y-3=\frac{3}{4}(x-1)[/tex]
Part 39) [tex]y+1=\frac{2}{3}(x-4)[/tex]
Part 45) The system has infinity solutions
Part 47) Is a exponential growth, the percent rate of change is 20%
Part 48) Is a exponential decay, the percent rate of change is -60%
Step-by-step explanation:
Part 38) Write an equation in point slope form of the line that pass through the given points
step 1
Find the slope
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
we have
(1,3) and (-3,0)
substitute
[tex]m=\frac{0-3}{-3-1}[/tex]
[tex]m=\frac{-3}{-4}[/tex]
[tex]m=\frac{3}{4}[/tex]
step 2
Find the equation in point slope form
[tex]y-y1=m(x-x1)[/tex]
we have
[tex]m=\frac{3}{4}[/tex]
[tex]point\ (1,3)[/tex]
substitute
[tex]y-3=\frac{3}{4}(x-1)[/tex]
Part 39) Write an equation in point slope form of the line that pass through the given points
step 1
Find the slope
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
we have
(-2,-5) and (4,-1)
substitute
[tex]m=\frac{-1+5}{4+2}[/tex]
[tex]m=\frac{4}{6}[/tex]
[tex]m=\frac{2}{3}[/tex]
step 2
Find the equation in point slope form
[tex]y-y1=m(x-x1)[/tex]
we have
[tex]m=\frac{2}{3}[/tex]
[tex]point\ (4,-1)[/tex]
substitute
[tex]y+1=\frac{2}{3}(x-4)[/tex]
Part 45) Solve the system of linear equations using any method
[tex]-5x-4y=-15[/tex] ----> equation A
[tex]10x+8y=30[/tex] ----> equation B
Multiply the equation A by -2 both sides
[tex]-2(-5x-4y)=-2(-15)[/tex]
[tex]10x+8y=30[/tex] -----> equation C
Compare equation B and equation C
Line B and Line C are the same line
so
The system has infinity solutions
Is a consistent dependent system
Part 47) Determine whether the function represent exponential growth or exponential decay. Identify the percent rate of change
we have
[tex]y=\frac{1}{4}(1.2)^t[/tex]
This is a exponential function of the form
[tex]y=a(b)^x[/tex]
where
a is the initial value or y-intercept
b is the base of the exponential function
r is the percent rate of change
b=(1+r)
If b < 1 ---> the function is a exponential decay
If b > 1 ---> the function is a exponential growth
In this problem we have
[tex]a=\frac{1}{4}[/tex]
[tex]b=1.2[/tex] ----> is a exponential growth
[tex]r=b-1=1.2-1=0.2[/tex]
convert to percentage
[tex]r=0.2*100=20\%[/tex]
Part 48) Determine whether the function represent exponential growth or exponential decay. Identify the percent rate of change
we have
[tex]f(t)=70(\frac{2}{5})^t[/tex]
This is a exponential function of the form
[tex]y=a(b)^x[/tex]
where
a is the initial value or y-intercept
b is the base of the exponential function
r is the percent rate of change
b=(1+r)
If b < 1 ---> the function is a exponential decay
If b > 1 ---> the function is a exponential growth
In this problem we have
[tex]a=70[/tex]
[tex]b=\frac{2}{5}=0.4[/tex] ----> is a exponential decay
[tex]r=b-1=0.4-1=-0.60[/tex] --->is negative because is a decreasing function
convert to percentage
[tex]r=-0.60*100=-60\%[/tex]
