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4 years ago

Instructions: Find the equation of the line through point (5, 1) and

Mathematics

1 answer:

marusya05 [52]4 years ago

7 0

make y the subject

y= 5x + 3

perpendicular line needs the reciprocal of the gradient

y = 1/5x + c

substitute in the point

1 = 1/5(5) + c

5/5 = 1 so the answer is y = 1/5x

hope this is helpful!

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What value of b will cause the system to have an infinite number of solutions?

irga5000 [103]

b must be equal to -6 for infinitely many solutions for system of equations $y = 6x + b$ and $-3 x+\frac{1}{2} y=-3$

<u>Solution: </u>

Need to calculate value of b so that given system of equations have an infinite number of solutions

$\begin{array}{l}{y=6 x+b} \\\\ {-3 x+\frac{1}{2} y=-3}\end{array}$

Let us bring the equations in same form for sake of simplicity in comparison

$\begin{array}{l}{y=6 x+b} \\\\ {\Rightarrow-6 x+y-b=0 \Rightarrow (1)} \\\\ {\Rightarrow-3 x+\frac{1}{2} y=-3} \\\\ {\Rightarrow -6 x+y=-6} \\\\ {\Rightarrow -6 x+y+6=0 \Rightarrow(2)}\end{array}$

Now we have two equations

$\begin{array}{l}{-6 x+y-b=0\Rightarrow(1)} \\\\ {-6 x+y+6=0\Rightarrow(2)}\end{array}$

Let us first see what is requirement for system of equations have an infinite number of solutions

If $a_{1} x+b_{1} y+c_{1}=0$ and $a_{2} x+b_{2} y+c_{2}=0$ are two equation

$\Rightarrow \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$ then the given system of equation has no infinitely many solutions.

In our case,

$\begin{array}{l}{a_{1}=-6, \mathrm{b}_{1}=1 \text { and } c_{1}=-\mathrm{b}} \\\\ {a_{2}=-6, \mathrm{b}_{2}=1 \text { and } c_{2}=6} \\\\ {\frac{a_{1}}{a_{2}}=\frac{-6}{-6}=1} \\\\ {\frac{b_{1}}{b_{2}}=\frac{1}{1}=1} \\\\ {\frac{c_{1}}{c_{2}}=\frac{-b}{6}}\end{array}$

As for infinitely many solutions $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$

$\begin{array}{l}{\Rightarrow 1=1=\frac{-b}{6}} \\\\ {\Rightarrow6=-b} \\\\ {\Rightarrow b=-6}\end{array}$

Hence b must be equal to -6 for infinitely many solutions for system of equations $y = 6x + b$ and $-3 x+\frac{1}{2} y=-3$

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3 years ago

A sinusoidal carrier wave has an amplitude of 5 volts, a frequency of 1MHz, and zero phase angle. Select the correct carrier wav

Vedmedyk [2.9K]

Answer:

The correct answer is b.

Step-by-step explanation:

The wave equation is given generally as:

c(x, t) = Acos(kx - wt)

Where A = amplitude

k = wave number

w = angular frequency.

x = horizontal distance moves by the wave.

t = time

The options show to us that the wave depends only on t and not (x, t).

Hence, the wave equation becomes:

c(t) = Acos(wt)

Given that:

A = 5 V

f = 1 * 10⁶ Hz

Angular Frequency, w, is given as:

w = 2πf

w = 2 * π * 1 * 10⁶ Hz

w = 2π(1 * 10⁶)

The wave equation becomes:

c(t) = 5cos(2*π*1*10⁶)

The correct answer is b.

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3 years ago