This document discusses different forms of linear functions and provides examples of determining the equation of a line given certain information. It introduces four forms of a linear function: slope-intercept form, standard form, point-slope form, and intercept form. It then gives three examples of finding the equation of a line using different forms, given the slope, two points on the line, or the x- and y-intercepts.

2. Recalling, a linear function comes in the form f(x) = mx +
b or y = mx + b. This form is referred to as the slope-intercept
form where m is the slope of the line and b represents the y-
intercept of the linear function.
We also have the standard form of linear function, which
is written in the form
ax + by = c
where a, b, and c are integers, and a > 0.

3. Recall the equation of the line using the following
forms:
1. Point-Slope Form 𝑦 − 𝑦1 = 𝑚 (𝑥 − 𝑥1)
2. Two-Point Slope 𝑦 − 𝑦1 =
𝑦2
−𝑦1
𝑥2
−𝑥1
(𝑥 − 𝑥1)
3. Slope-Intercept Form 𝑦 = 𝑚𝑥 + 𝑏
4. Intercept Form
𝑥
𝑎
+
𝑦
𝑏
= 1

4. Example1.
Find the general term of the equation of the line which
through P1(-1, 8) and whose slope is m =
𝟐
𝟑
.
Steps Solution
1. Identify the given information
m =
𝟐
𝟑
and P1(-1, 8)
x1 = -1 and y1 = 8
2. Identify the formula to be
used.
𝑦 − 𝑦1 = 𝑚 (𝑥 − 𝑥1)

5. Steps Solution
3. Substitute the given
information in step 1 to the
formula in step 2.
𝑦 − 𝑦1 = 𝑚 (𝑥 − 𝑥1)
𝑦 − 8 =
𝟐
𝟑
𝑥 − (−1)
4. Find the general equation of
the line
ax + by + c = 0
𝑦 − 8 =
𝟐
𝟑
(𝑥 + 1)
3y – 24 = 2(x + 1)
3y – 24 = 2x + 2
-2x + 3y -24 – 2 = 0
-2x + 3y - 26 = 0
(-2x + 3y - 26 )(-1)= 0(-1)
2x – 3y + 26 = 0
Thus, the general form of the equation of the line is 2x – 3y + 26 = 0.

6. Example 2.
Determine the general form of the equation of the line
passing through P1(5, -3) and P2(2, 7).
Steps Solution
1. Identify the given information P1(5, -3) and P2(2, 7)
x1= 5, y1 = -3, x2 = 2 and y2 = 7
2. Identify the formula to be
used.
𝑦 − 𝑦1 =
𝑦2
−𝑦1
𝑥2
−𝑥1
(𝑥 − 𝑥1)

7. Steps Solution
3. Substitute the given
information in step 1 to the
formula in step 2.
𝑦 − 𝑦1 =
𝑦2
−𝑦1
𝑥2
−𝑥1
(𝑥 − 𝑥1)
y – (-3) =
7 −(−3)
2 −5
(x – 5)
4. Find the general equation
the line.
y + 3 =
7+3
−3
(x – 5)
y + 3 =
10
−3
(x – 5)
y + 3 =
10
−3
(x – 5) (-3)
-3y – 9 = 10(x – 5)
-3y – 9 = 10x – 50
-10x – 3y – 9 + 50 = 0
−10x – 3y + 41 (-1)= 0(-1)
10x + 3y – 41 = 0
Therefore the general form of the equation is 10x + 3y – 41 = 0

8. Example 3.
Determine the slope-intercept form of the equation of the
line whose x – intercept and y-intercept are (-5, 0) and (0, 4),
respectively.
Steps Solution
1. Identify the given.
We have (-5, 0) and (0, 4)
a = -5 and b = 4
2. Identify the formula.
𝑥
𝑎
+
𝑦
𝑏
= 1
3. Find the slope-intercept form
𝑥
−5
+
𝑦
4
= 1 (20)
4x + 5y = 20
5y = - 4x + 20
y = -
𝟒
𝟓
x + 4