Download Writing a Linear Equation

Writing a Linear Equation
The ability to write a linear equation is an important skill, as it
gives us a way to represent a line into a single statement.
If we are given the slope and y-intercept of a line, we can easily
find its equation by using the slope-intercept form of an
equation.
Slope-Intercept Form: y = mx + b
where m is the slope and (0, b) is the y-intercept
Example 1: Write the equation of the line with slope -4 and point (0, 7).
y = mx + b
y = -4x + 7
Example 2: Write the equation of the line with slope 9 and point (0, -1).
y = mx + b
y = 9x – 1
Example 3: Write the equation of the line with slope
and point (0, 6).
y = mx + b
y=
x+6
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
The slope-intercept form has the advantage of being simple to
remember and use, however, we must know the y-intercept. If
we know the slope of an equation and any point (x1, y1) on the
line, we can use the point-slope formula.
Point-Slope Formula: y – y1 = m (x – x1)
where m is the slope and (x1, y1) is any point on the line
Example 4: Write the equation of the line with slope 2 and point (3, -4).
y – y1 = m (x – x1)
y – (-4) = 2 (x – 3)
y + 4 = 2 (x – 3)
Simplify the double negative
y + 4 = 2x – 6
Distribute the 2
-4
-4
Subtract 4 from both sides
y = 2x – 10
Example 5: Write the equation of the line with slope -3 and point (-5, 0)
y – y1 = m (x – x1)
y – 0 = -3 (x – (-5))
y – 0 = -3 (x + 5)
Simplify the double negative
y = -3 (x + 5)
“Subtract” 0
y = -3x – 15
Distribute the -3
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 6: Write the equation of the line with slope - and point (6, 2)
y – y1 = m (x – x1)
y – 2 = - (x – 6)
y–2=- x+4
+2
Distribute the -
+2
Add 2 to both sides
y=- x+6
In order to write an equation of a line, we will always need to
know the slope. If we are given two points, we can use the
formula for finding slope and then use one of the given points.
Example 7: Write the equation of the line with points (-2, 5) and (4, -3).
m=
=
=
=
First find the slope
y – y1 = m (x – x1)
y–5=
(x – (-2))
Use either point
y–5=
(x + 2)
Simplify the double-negative
y–5=
x–
Distribute the
+5
y=
+5
Add 5 to both sides
x+
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 8: Write the equation of the line with points (-3, 4) and (-1, -2)
m=
=
y – y1 = m (x – x1)
y – 4 = -3 (x – (-3))
y – 4 = -3 (x + 3)
y – 4 = -3x – 9
+4
+4
y = -3x – 5
=
= -3
First find the slope
Simplify the double-negative
Distribute the -3
Add 4 to both sides
Recall that horizontal lines have a slope of 0, and vertical lines
have an undefined line. Writing the equations of horizontal and
vertical lines is quite simple.
Example 9: Write the equation of the line with slope 0 and point (3, 8)
y – y1 = m (x – x1)
y – 8 = 0 (x – 3)
y–8=0
Multiply by 0
+8+8
Add 8 to both sides
y=8
With the above example, we can conclude that the equation for a
horizontal line will always be y = the y-value of the point (all
the y-values of every point on the line will be the same).
Similarly, the equation for a vertical line will always be x = the
x-value of the point (all the x-values of every point on the line
will be the same).
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 10: Write the equation of the line with an undefined slope and
point (-2, -5).
Since the x-value = -5, the equation is x = -5
Example 11: Write the equation of the line with slope 0 and point (1, -3)
Since the y-value = -3, the equation is y = -3
Recall that when we are given a linear equation, we can find the
slope of a line that is parallel or perpendicular.
Parallel lines will have the same slope
Perpendicular lines will have slopes that are opposite reciprocals
Once we have a slope, it is possible to write the equation of the
line that is parallel or perpendicular if given one of its points.
Example 12: Write the equation for a line parallel to y = 2x + 4 and
passing through the point (4, -3).
y = 2x + 4 has a slope of 2, so we need a slope of 2
y – y1 = m (x – x1)
y – (-3) = 2 (x – 4)
y + 3 = 2 (x – 4)
y + 3 = 2x – 8
-3
-3
y = 2x – 11
Simplify the double-negative
Distribute the 2
Subtract 3 from both sides
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 13: Write the equation for a line perpendicular to y = - x + 4
and passing through the point (6, -9).
y = - x + 4 has a slope of - , so we need a slope of
y – y1 = m (x – x1)
y – (-9) =
(x – 6)
y+9=
(x – 6)
y + 9 = x – 10
-9
-9
Simplify the double-negative
Distribute the
Subtract 9 from both sides
y = x – 19
Example 14: Write the equation for a line perpendicular to x = -2
and passing through the point (3, 4).
x = -2 is a vertical line with an undefined slope
a perpendicular line would be a horizontal line with slope 0
y – y1 = m (x – x1)
y – 4 = 0 (x – 3)
y–4=0
Multiply by 0
+4+4
Add 4 to both sides
y=4
Which is a horizontal line
Note that we didn’t necessarily need to use the point-slope
formula. We could have considered the given point (3, 4)
and simply written the equation y = 4.
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 15: Write the equation for a line parallel to 2x – 3y = 6
and passing through the point (3, 5).
2x – 3y = 6
- 2x
- 2x
- 3y = -2x + 6
-3
-3
-3
First find the slope
Subtract 2x from both sides
Divide each term by -3
y = x -2
The slope is , so we need a slope of
y – y1 = m (x – x1)
y–5=
(x – 3)
y–5= x–2
+5
+5
Distribute the
Add 5 to both sides
y= x+3
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)