SLOPE of a LINEAR EQUATION, Ax + by = C

SLOPE OF A LINEAR EQUATION, Ax + By = C

The slope of a line is a ratio of the vertical change to the horizontal change between any two points on a line.

Slope is denoted as "m".

Slope can be referred to in the following ways given below.

slope = vertical change = change in y = Δ y = rise
horizontal change change in x Δ x run

FINDING SLOPE WHEN GIVEN TWO POINTS, (x1, y1) and (x2, y2):
The slope, m, of any two points, (x1, y1) and (x2, y2), on a line is defined by
slope = m = or

EXAMPLES:

1) Find the slope of a line that passes through the points (5, 9) and (-2, 4).

m = 9 - 4 = 5 is the slope.
5 - (-2) 7

Note: if you set up the formula as such, 4 - 9 = - 5 = 5 , the slope is still 5.
-2 - 5 -7 7 7

2 ) Find the slope of a line that passes through the points, and

m = = = is the solution .

Note: Complex fractions should be reduce to simple fractions for the final slope solution
To simplify: numerator ÷ denominator

3) Find the slope of aline that passes through the points (3, 7) and (3, -10).

m = 7 - (-10) = 17 . The slope is undefined.
3 - 3 0
NOTE: When the slope is undefined, then the graph of the line that passes through those points is a vertical line.

4) Find the slope of a line that passes through the points (5, -4) and (6, -4).

m = -4 - (-4) = 0 = 0. The slope is zero.
5 - 6 - 1
NOTE: When the slope is zero, then the graph of the line that passes through those points is a horizontal line.

FINDING SLOPE FROM THE EQUATION OF A LINE:

Definition: y = mx + b is the slope-intercept form of the linear equation.
where m is the slope. (m is the coefficient of x)
b is the y-intercept, (0, b). (b is the constant)
(x, y) are the ordered pair values.

STEPS:

1) Convert the equation into slope-intercept form, if necessary, by solving for
"y" completely.
2) Identify the slope. (m = the coefficient of x)

EXAMPLES:

1) Find the slope in the linear equation, y = 3x - 5.

Since the equation of the line is in slope-intercept form, then m = 3.

2) Find the slope in the linear equation, 5x - 12y = 10.

Solve for y: 5x – 12y = 10
– 12y = 10 – 5x
is in slope-intercept form.

Identify the slope: The slope is .

3) Find the slope in the linear equation, 2(x - 4) - (3 - y) = 4x + 5y - 7.
Solve for y: 2x – 8 – 3 + y = 4x + 5y – 7
2x – 11 + y = 4x + 5y – 7
2x – 11 – 4y = 4x – 7
– 4y = 2x + 4
is in slope-intercept form.

Identify the slope: The slope is .

4) Find the slope in the linear equation, 3y + 9 = 15.

Solve for y: 3y = 6
y = 2 is in slope-intercept form.

Identify the slope:
There is no "x" variable. However, it is understood to be there when we rewrite the equation as
y = 2 + 0x. Therefore, the slope is zero and we know the line is horizontal.

5) Find the slope in the linear equation, 5x - 2y = 10 - 2y.
Solve for y: 5x = 10
x = 2

Identify the slope:
There is no "y" variable. However, it is understood to be there when we rewrite the equation as 0y + x = 2. When we solve for y, the equation is and the slope is which is undefined. Therefore, when the equation does not have a "y" variable to solve for, the slope is undefined and the line will be vertical.

SLOPES OF PARALLEL AND PERPENDICULAR LINES

DEFINITIONS:
Two lines are parallel if the two lines never intersect.
Two lines are perpendicular if the two lines form a right angle (90 degrees).

THEOREM:

1) If two lines are parallel, then they have equal slopes (m1 = m2).

Example: L1: 2x - 3y = 14 (y = 2/3x - 14/3) then m = 2/3
L2: y = 2/3x + 5 then m = 2/3

Lines L1 and L2 are parallel.

2) If two lines are perpendicular, then the slopes are negative reciprocals,
(m1∙ m2 = -1).

Example: L1: 2x - 3y = 1 (y = 2/3x - 1/3) then m = 2/3
L2: y = -3/2x + 6 then m = -3/2

to check if slopes are negative reciprocals, 2/3 ∙ -3/2 = -1.
Lines L1 and L2 are perpendicular.