Linear equations are the foundation of algebraic graphing. They describe straight lines and are crucial for modeling relationships between variables. Understanding how to create and manipulate these equations is key to solving real-world problems and more complex math.
This section covers different methods for writing linear equations. You'll learn how to use , intercepts, and points to construct equations. These skills are essential for graphing lines and analyzing their relationships, setting the stage for more advanced algebraic concepts.
Equations of Lines
Line equation from slope and y-intercept
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[y = mx + b](https://www.fiveableKeyTerm:y_=_mx_+_b) used to construct line equation
m slope of the line indicates steepness and direction (positive slopes rise, negative slopes fall)
[b](https://www.fiveableKeyTerm:b) where line crosses (0, b) provides starting point
Substitute given slope value for m and y-intercept value for b
Plug in values to create specific equation for the line
Example: slope 2, y-intercept -3
y=2x−3 is the equation of the line
Line equation from slope and point
y−y1=m(x−x1) derives line equation
m slope of the line indicates steepness and direction
(x1,y1) coordinates of a point on the line serve as a reference
Substitute given slope for m and point coordinates for x1 and y1
Plug in values to create specific equation for the line
Simplify equation by distributing slope and combining like terms
Example: slope -1/2, point (4, 6)
y−6=−21(x−4) is the equation of the line
Line equation from two points
Calculate slope m using slope formula m=x2−x1y2−y1
(x1,y1) and (x2,y2) are the coordinates of the two given points
Slope represents change in y over change in x ()
Substitute calculated slope m and coordinates of either point into point-slope form y−y1=m(x−x1)
Plug in values to create specific equation for the line
Simplify equation by distributing slope and combining like terms
Example: points (1, 3) and (5, 11)
m=5−111−3=2
y−3=2(x−1) is the equation of the line
Equation of parallel line
lines have the same slope but different y-intercepts
Identical steepness and direction but shifted up or down
Use the same slope as the given line and a different y-intercept or point to create parallel line equation
Maintain slope but adjust the starting point
Substitute slope and y-intercept or point into slope-intercept form y=mx+b or point-slope form y−y1=m(x−x1)
Example: given line y=3x−2, parallel line through point (1, 5)
y−5=3(x−1) is the equation of the parallel line
Equation of perpendicular line
lines have slopes that are negative reciprocals m1=−m21
Product of slopes equals -1, indicating 90° angle between lines
Find negative reciprocal of given line's slope
If given slope is m, perpendicular slope is −m1
Use negative reciprocal slope and given point to create perpendicular line equation with point-slope form y−y1=m(x−x1)
Example: given line y=−2x+3, perpendicular line through point (2, -1)
mperp=21 is the slope of the perpendicular line
y−(−1)=21(x−2) is the equation of the perpendicular line
Linear Functions and Graphs
A is a relationship between variables that produces a straight line when graphed on a
The in a linear function is constant, represented by the slope
The is a visual representation of all points satisfying a
Linear equations can be written in various forms, including slope-intercept and point-slope