Linear Equations
Slope-Intercept Form | Point-Slope Form |
Horizontal & Vertical Lines
Linear Equations
Learning Objectives
• Determine the equation of a line given the slope
and y-intercept
• Determine the equation of a line given the slope
and any point
• Determine the equation of a line given any two
points on the line
Slope-Intercept Form
• Standard-form – an equation of a line in the form
Ax + By = C
– Value of A is greater than or equal to 0
– Value of A and B are not both 0
– Value of A, B, and C are real number constants
• Ex) 3x + 2y = 6
– Subtract 3x from both sides of equation 2y = –3x + 6
– Then, divide both sides of equation by 2 for
-3x
y=
+3
2
Slope-Intercept Form
• Lines can be graphed
by identifying the
transformations on the
linear parent function,
y=x
-3x
– Ex) y = 2 + 3
• Reflected over the y-axis
• Vertically stretched
• Vertically translated up
the y-axis by 3
• (0,3) is the y-intercept
• Slope is –3/2
Slope-Intercept Form
Slope-intercept form of a line
The slope-intercept form of an equation of a line is in the form
y = mx + b
where m is the slope of the line, and b is the y-intercept.
• Equation can be written quickly if slope and
y-intercept are known
• Slope and y-intercept recognized immediately in
equation and line can be graphed quickly
Point-Slope Form
• When a slope is given without a y-intercept, an
equation can be formed from a point on the line
– Must apply transformations to the linear parent function
until the equation represents the desired line
• Ex) If slope is 1/2, then the graph of the linear parent function
should be vertically compressed
1
y= x
2
• If the point on the line is (4,5), then translate the point to the right
by four units and up by five units
1
y = ( x - 4) + 5
2
• Value of y for the given point can be moved to the other side of
the equation
1
y - 5 = ( x - 4)
2
Point-Slope Form
Point-slope form of a line
The point-slope form of the equation of a line is
y – y1 = m(x – x1)
where m is the slope of the line, and (x1,y1) is a point on the line.
• Advantages of point-slope form
– Equation can be formed quickly from the slope and one
point of a line
– Can find y-intercept by solving the equation for y and
converting the equation into slope-intercept form
Point-Slope Form Example
Ex) A line passes through the following two points, (1,4)
and (5,7). Find the equation of this line in point–slope
form and slope–intercept form.
Analyze
Formulate
Justify
Find slope and substitute into
point–slope equation, solve for y
Determine
y1 - y2 7 - 4 3
=
=
x1 - x2 5 - 1 4
y=
3
( x - 1) + 4
4
y - y1 = m( x - x1 )
y=
3
3
x- +4
4
4
3
y - 4 = ( x - 1)
4
y=
3
1
x+3
4
4
Slope is located to the left of
the x-value in each form
Evaluate
Depending on the information
given, certain forms can be
created faster
Horizontal & Vertical Lines
• Sometimes linear equations do not appear to
contain both x- and y-components
– Ex) y = 4
• Two points on the line are
(–3,4) and (5,4)
• Slope is
y1 - y2
4-4
0
=
= =0
x1 - x2 5 - ( -3 ) 8
• Standard form is 0x + y = 4
• Slope–intercept form is
y = 0x + 4
• Possible point–slope equation
is y – 4 = 0(x – 5)
Horizontal & Vertical Lines
• Ex) x = 4
– Two points on the line
are (4, –2) and (4,3)
– Slope is
y1 - y2 3 - ( -2) 5
=
= = undefined
x1 - x2
4-4
0
– Impossible to write
equation of a vertical
line in slope–intercept
or point–slope form
– Standard form is
x + 0y = 4
Linear Equations
Learning Objectives
• Determine the equation of a line given the slope
and y-intercept
• Determine the equation of a line given the slope
and any point
• Determine the equation of a line given any two
points on the line