Equation of a Line: Given Slope and One Point

Equation of a Line: Given Slope and One Point

Example 1: Determine the equation of a line with a slope of 2 passing through the point (1, 4).

Solution: m=2, (-1,4)

Write the equation of the line in slope y-intercept form,

y=mx+b Write the form you’re using.

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y=2x+b Substitute the slope into m.

-Sub. x = – 1 and y = 4 into (*) and solve for b.

y = 2x + b

4 = 2(- 1) + b Substitute the coordinates into x and y.

4 = – 2 + b Simplify.

4+2 = b Isolate the ‘b’ term

6 = b Simplify

b = 6 Rearrange so the variable is on the left.

-Now substitute m = 2 and b = 6 in y = mx+b.

∴y = 2x + 6 This is the equation of the line with a slope of 2 possing through the point (-1,4).

Example 2: Determine the equation of the line with a slope of 2/3 passing through (6, 7).

Solution: m=23, (6, -7)

y = mx + b ← Write the form you’re using.

y = 23x + b ← Substitute the slope into m.

– 7 = 23(6) + b ← Substitute the coordinates into x and y.

– 7 = (2)(2) + b ← Simplify.

– 7 = 4 + b ← Isolate the ‘b’ term

– 7 – 4 =b ← Simplify

– 11 = b ← Simplify

b = – 11 Rearrange so the variable is on the left.

-Sub. m=23 and b = – 11 into y= mx+b

Write the equation

∴ y=23 x – 11

Equation of the line that has a slope of 23 and passes through

(6,-7).

Example 3: Find the equation of the line parallel to y = 3x 2 passing through the point (2, 5).

Solution: y = 3x – 2, m = 3

-Since the line whose equation we are trying to write is parallel to y = 3x – 2, then the slope of this line is 3.

-Use m = 3 and (-2, 5) to write the equation of the line.

y = mx + b y= 3x +11

Y = 3x + b

-Use (-2, 5) to find b

5 = 3(- 2) + b

5 = – 6 + b

5+6 = b

11 = b

b=11

– Sub . m = 3 and b = 11 into y = mx + b

∴ y = 3x + 11 Equation of the line that is parallel to y = 3x – 2 passing through (-2, 5)

Example 4: Find the equation of the line perpendicular to 2x-3y + 7 = 0 with the same yintercept as 4x + 5y 15 = 0.

Solution:

Write 2x-3y +7=0 in slope y-intercept form, y = mx + b

2x + 7 = 3y

3y = 2x + 7

3y3=2x+73

y=23x+73

So, m=23

The slope of the line perpendicular to

2x – 3y + 7 = 0 is -32 ( recall : m1=-1m2)

-Next, find the y-intercept of 4x+5y-15=0

Set x=0,

As a point (0,3)

-Write the equation of the line using the slope of -32 and point (0,3).

∴ y=23 x +3 is the equation of the – line that is perpendicular to 2x – 3y + 7 = 3 with the same y-intercept as 4x+5y-15=0