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2.1 Linear functions  (Page 5/17)

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Therefore, the same line can be described in slope-intercept form as y = − 1 2 x + 7.

Point-slope form of a linear equation

The point-slope form of a linear equation takes the form

y − y 1 = m ( x − x 1 )

where m is the slope, x 1   and   y 1 are the x  and  y coordinates of a specific point through which the line passes.

Writing the equation of a line using a point and the slope

The point-slope form is particularly useful if we know one point and the slope of a line. Suppose, for example, we are told that a line has a slope of 2 and passes through the point ( 4 , 1 ) . We know that m = 2 and that x 1 = 4 and y 1 = 1. We can substitute these values into the general point-slope equation.

y − y 1 = m ( x − x 1 ) y − 1 = 2 ( x − 4 )

If we wanted to then rewrite the equation in slope-intercept form, we apply algebraic techniques.

y − 1 = 2 ( x − 4 ) y − 1 = 2 x − 8 Distribute the  2.         y = 2 x − 7 Add 1 to each side .

Both equations, y − 1 = 2 ( x − 4 ) and y = 2 x – 7 , describe the same line. See [link] .

Writing linear equations using a point and the slope

Write the point-slope form of an equation of a line with a slope of 3 that passes through the point ( 6 , –1 ) . Then rewrite it in the slope-intercept form.

Let’s figure out what we know from the given information. The slope is 3, so m = 3. We also know one point, so we know x 1 = 6 and y 1 = −1. Now we can substitute these values into the general point-slope equation.

       y − y 1 = m ( x − x 1 ) y − ( − 1 ) = 3 ( x − 6 ) Substitute known values .          y + 1 = 3 ( x − 6 ) Distribute  − 1  to find point-slope form .

Then we use algebra to find the slope-intercept form.

y + 1 = 3 ( x − 6 ) y + 1 = 3 x − 18 Distribute 3 .         y = 3 x − 19 Simplify to slope-intercept form .
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Write the point-slope form of an equation of a line with a slope of –2 that passes through the point ( –2 ,   2 ) . Then rewrite it in the slope-intercept form.

y − 2 = − 2 ( x + 2 ) ; y = − 2 x − 2

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Writing the equation of a line using two points

The point-slope form of an equation is also useful if we know any two points through which a line passes. Suppose, for example, we know that a line passes through the points ( 0 ,   1 ) and ( 3 ,   2 ) . We can use the coordinates of the two points to find the slope.

m = y 2 − y 1 x 2 − x 1     = 2 − 1 3 − 0     = 1 3

Now we can use the slope we found and the coordinates of one of the points to find the equation for the line. Let use (0, 1) for our point.

y − y 1 = m ( x − x 1 ) y − 1 = 1 3 ( x − 0 )

As before, we can use algebra to rewrite the equation in the slope-intercept form.

y − 1 = 1 3 ( x − 0 ) y − 1 = 1 3 x Distribute the  1 3 .        y = 1 3 x + 1 Add 1 to each side .

Both equations describe the line shown in [link] .

Writing linear equations using two points

Write the point-slope form of an equation of a line that passes through the points (5, 1) and (8, 7). Then rewrite it in the slope-intercept form.

Let’s begin by finding the slope.

m = y 2 − y 1 x 2 − x 1    = 7 − 1 8 − 5    = 6 3    = 2

So m = 2. Next, we substitute the slope and the coordinates for one of the points into the general point-slope equation. We can choose either point, but we will use ( 5 , 1 ) .

y − y 1 = m ( x − x 1 ) y − 1 = 2 ( x − 5 )

The point-slope equation of the line is y 2 – 1 = 2 ( x 2 – 5 ) . To rewrite the equation in slope-intercept form, we use algebra.

y − 1 = 2 ( x − 5 ) y − 1 = 2 x − 10        y = 2 x − 9

The slope-intercept equation of the line is y = 2 x – 9.

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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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