1. Writing Linear Equations
Linear equations can be represented in three ways :
1. Standard Form 𝑎𝑥+𝑏𝑦=𝑐
- where 𝑎 and 𝑏 are not both zero

2. Writing Linear Equations
Linear equations can be represented in three ways :
1. Standard Form 𝑎𝑥+𝑏𝑦=𝑐
- where 𝑎 and 𝑏 are not both zero
An example would be 4𝑥−3𝑦= 𝑎=4 , 𝑏=−3 , 𝑐=12

3. Writing Linear Equations
Linear equations can be represented in three ways :
1. Standard Form 𝑎𝑥+𝑏𝑦=𝑐
- where 𝑎 and 𝑏 are not both zero
An example would be 4𝑥−3𝑦= 𝑎=4 , 𝑏=−3 , 𝑐=12
2. Point – Slope Form 𝑦−𝑏=𝑚(𝑥−𝑎)
- where 𝑚 is the slope, and ( 𝑎 , 𝑏 ) is a point on the line

4. Writing Linear Equations
Linear equations can be represented in three ways :
1. Standard Form 𝑎𝑥+𝑏𝑦=𝑐
- where 𝑎 and 𝑏 are not both zero
An example would be 4𝑥−3𝑦= 𝑎=4 , 𝑏=−3 , 𝑐=12
2. Point – Slope Form 𝑦−𝑏=𝑚(𝑥−𝑎)
- where 𝑚 is the slope, and ( 𝑎 , 𝑏 ) is a point on the line
An example would be 𝑦−2=−2(𝑥+5)
𝑚=−2, point (2 , −5 )

5. Writing Linear Equations
Linear equations can be represented in three ways :
1. Standard Form 𝑎𝑥+𝑏𝑦=𝑐
- where 𝑎 and 𝑏 are not both zero
An example would be 4𝑥−3𝑦= 𝑎=4 , 𝑏=−3 , 𝑐=12
2. Point – Slope Form 𝑦−𝑏=𝑚(𝑥−𝑎)
- where 𝑚 is the slope, and ( 𝑎 , 𝑏 ) is a point on the line
An example would be 𝑦−2=−2(𝑥+5)
𝑚=−2, point (2 , −5 )
3. Slope – intercept Form 𝑦=𝑚𝑥+𝑏
- where 𝑚 is the slope, and 0 , 𝑏 is the y - intercept

6. Writing Linear Equations
Linear equations can be represented in three ways :
1. Standard Form 𝑎𝑥+𝑏𝑦=𝑐
- where 𝑎 and 𝑏 are not both zero
An example would be 4𝑥−3𝑦= 𝑎=4 , 𝑏=−3 , 𝑐=12
2. Point – Slope Form 𝑦−𝑏=𝑚(𝑥−𝑎)
- where 𝑚 is the slope, and ( 𝑎 , 𝑏 ) is a point on the line
An example would be 𝑦−2=−2(𝑥+5)
𝑚=−2, point (2 , −5 )
3. Slope – intercept Form 𝑦=𝑚𝑥+𝑏
- where 𝑚 is the slope, and 0 , 𝑏 is the y – intercept
An example would be 𝑦=3𝑥+5
𝑚=3 , y – intercept is ( 0 , 5 )

7. Writing Linear Equations
Standard Form and Point – Slope Form can both be written as Slope – Intercept Form with a little Algebra. Slope – Intercept Form is probably the most widely used format for linear equations / functions.

8. Writing Linear Equations
Standard Form and Point – Slope Form can both be written as Slope – Intercept Form with a little Algebra. Slope – Intercept Form is probably the most widely used format for linear equations / functions.
EXAMPLE : Find the slope and y – intercept for 2𝑥+3𝑦=−15

9. Writing Linear Equations
Standard Form and Point – Slope Form can both be written as Slope – Intercept Form with a little Algebra. Slope – Intercept Form is probably the most widely used format for linear equations / functions.
EXAMPLE : Find the slope and y – intercept for 2𝑥+3𝑦=−15
To write this equation in slope – intercept form, solve for 𝑦…

10. Writing Linear Equations
Standard Form and Point – Slope Form can both be written as Slope – Intercept Form with a little Algebra. Slope – Intercept Form is probably the most widely used format for linear equations / functions.
EXAMPLE : Find the slope and y – intercept for 2𝑥+3𝑦=−15
To write this equation in slope – intercept form, solve for 𝑦…
2𝑥+3𝑦=−15
−2𝑥 =−2𝑥
3𝑦=−2𝑥−15

11. Writing Linear Equations
Standard Form and Point – Slope Form can both be written as Slope – Intercept Form with a little Algebra. Slope – Intercept Form is probably the most widely used format for linear equations / functions.
EXAMPLE : Find the slope and y – intercept for 2𝑥+3𝑦=−15
To write this equation in slope – intercept form, solve for 𝑦…
2𝑥+3𝑦=−15
−2𝑥 =−2𝑥
3𝑦=−2𝑥−15
3𝑦 3 = −2𝑥 3 − 15 3

12. Writing Linear Equations
Standard Form and Point – Slope Form can both be written as Slope – Intercept Form with a little Algebra. Slope – Intercept Form is probably the most widely used format for linear equations / functions.
EXAMPLE : Find the slope and y – intercept for 2𝑥+3𝑦=−15
To write this equation in slope – intercept form, solve for 𝑦…
2𝑥+3𝑦=−15
−2𝑥 =−2𝑥
3𝑦=−2𝑥−15
3𝑦 3 = −2𝑥 3 − 15 3
𝑦= −2 3 𝑥−5

13. Writing Linear Equations
Standard Form and Point – Slope Form can both be written as Slope – Intercept Form with a little Algebra. Slope – Intercept Form is probably the most widely used format for linear equations / functions.
EXAMPLE : Find the slope and y – intercept for 2𝑥+3𝑦=−15
To write this equation in slope – intercept form, solve for 𝑦…
2𝑥+3𝑦=−15
−2𝑥 =−2𝑥
3𝑦=−2𝑥−15
3𝑦 3 = −2𝑥 3 − 15 3
𝑦= −2 3 𝑥−5
𝑚= −2 3 , y – intercept =( 0 , −5)

14. Writing Linear Equations
EXAMPLE #2 : What is the slope and y – intercept for 𝑦−3=4(𝑥−2)

15. Writing Linear Equations
EXAMPLE #2 : What is the slope and y – intercept for 𝑦−3=4(𝑥−2)
Now, technically speaking, there is no Algebra needed. The slope is given in the format, 𝑚=4.

16. Writing Linear Equations
EXAMPLE #2 : What is the slope and y – intercept for 𝑦−3=4(𝑥−2)
Now, technically speaking, there is no Algebra needed. The slope is given in the format, 𝑚=4.
And since the y – intercept always has an x – value of zero, we can substitute
𝑥=0 and solve for 𝑦.

17. Writing Linear Equations
EXAMPLE #2 : What is the slope and y – intercept for 𝑦−3=4(𝑥−2)
Now, technically speaking, there is no Algebra needed. The slope is given in the format, 𝑚=4.
And since the y – intercept always has an x – value of zero, we can substitute
𝑥=0 and solve for 𝑦.
𝑦−3=4(0−2)
𝑦−3=4 −2
𝑦−3=−8
+3 =+3
𝑦=−5 y – intercept ( 0 , −5 )

18. Writing Linear Equations
EXAMPLE #3 : What is the x – intercept for 3𝑥−5𝑦=−15

19. Writing Linear Equations
EXAMPLE #3 : What is the x – intercept for 3𝑥−5𝑦=−15
x – intercepts occur where 𝑦=0. Substitute and solve for 𝑥.

20. Writing Linear Equations
EXAMPLE #3 : What is the x – intercept for 3𝑥−5𝑦=−15
x – intercepts occur where 𝑦=0. Substitute and solve for 𝑥.
3𝑥−5 0 =−15
3𝑥=−15
3𝑥 3 = −15 3
𝑥=− x – intercept is ( −5 , 0 )

21. Writing Linear Equations
How do we find the equation of a line given a graph ? It depends what you are given as your graph to work from.

22. Writing Linear Equations
How do we find the equation of a line given a graph ? It depends what you are given as your graph to work from.
This graph has a clear y – intercept and I can see the slope by counting using points.

23. Writing Linear Equations
How do we find the equation of a line given a graph ? It depends what you are given as your graph to work from.
This graph has a clear y – intercept and I can see the slope by counting using points.
Y intercept is ( 0 , 4 )
𝑚= 3 2 +2 +3
Slope – intercept form :
𝑦= 3 2 𝑥+4

24. Writing Linear Equations
This graph does not have a clear y – intercept. But I can still identify some points, find the slope, and then use one of the points as 𝑎 , 𝑏 in point – slope form.

25. Writing Linear Equations
This graph does not have a clear y – intercept. But I can still identify some points, find the slope, and then use one of the points as 𝑎 , 𝑏 in point – slope form.
𝑚= −4−(−1) 6−(−1) = −3 7
Using the point ( 6 , −4 ) and point – slope form :
𝑦− −4 = 3 7 𝑥−6 OR
𝑦+4= 3 7 (𝑥−6)
( −𝟏 , −𝟏 )
( 𝟔 , −𝟒 )

26. Writing Linear Equations
Find the equation of a line in slope - intercept form given two points on the line.
Given points 2 , −1 , ( 3 , 3)

27. Writing Linear Equations
Find the equation of a line in slope - intercept form given two points on the line.
Given points 2 , −1 , 3 , 3
1. Find the slope 𝑚= 3−(−1) 3−2) = 4 1

28. Writing Linear Equations
Find the equation of a line in slope - intercept form given two points on the line.
Given points 2 , −1 , 3 , 3
Find the slope 𝑚= 3−(−1) 3−2) = 4 1
Use the point – slope form with 𝑚=4 and the point ( 3 , 3 )
𝑦−3=4(𝑥−3)

29. Writing Linear Equations
Find the equation of a line in slope - intercept form given two points on the line.
Given points 2 , −1 , 3 , 3
Find the slope 𝑚= 3−(−1) 3−2) = 4 1
Use the point – slope form with 𝑚=4 and the point ( 3 , 3 )
𝑦−3=4(𝑥−3)
Solve for 𝑦
𝑦−3=4𝑥−12
+3=
𝑦=4𝑥−9
𝑚=4 , y – intercept is ( 0 , −9 )

30. Writing Linear Equations
EXAMPLE : The temperature of a liquid cools 4° F every minute. After 5 minutes, the temperature of the liquid is 175° F. Which equation best models the temperature, 𝑦, of this liquid after 𝑥 minutes ?
A) 𝑦=−4𝑥+155 C) 𝑦=−5𝑥+155
B) 𝑦=−4𝑥+195 D) 𝑦=−5𝑥+195

31. Writing Linear Equations
EXAMPLE : The temperature of a liquid cools 4° F every minute. After 5 minutes, the temperature of the liquid is 175° F. Which equation best models the temperature, 𝑦, of this liquid after 𝑥 minutes ?
A) 𝑦=−4𝑥+155 C) 𝑦=−5𝑥+155
B) 𝑦=−4𝑥+195 D) 𝑦=−5𝑥+195
SOLUTION : The slope 𝑚=−4 because of the liquid cooling 4° F every minute.
It is a rate of change…

32. Writing Linear Equations
EXAMPLE : The temperature of a liquid cools 4° F every minute. After 5 minutes, the temperature of the liquid is 175° F. Which equation best models the temperature, 𝑦, of this liquid after 𝑥 minutes ?
A) 𝑦=−4𝑥+155 C) 𝑦=−5𝑥+155
B) 𝑦=−4𝑥+195 D) 𝑦=−5𝑥+195
SOLUTION : The slope 𝑚=−4 because of the liquid cooling 4° F every minute.
It is a rate of change…
We can use the point ( 5 , 175 ) as a point on the line.
𝑦−𝑏=𝑚 𝑥−𝑎
𝑦−175=−4(𝑥−5)
𝑦−175=−4𝑥+20
+175=
𝑦=−4𝑥+195
So ANSWER is B