The relative position of two straight lines can be determined from a graph or an equation. Consider the following cases:
| Distinct Parallel Lines (same slope, but different y-intercept) |
Coincident Parallel Lines (same slope and same y-intercept) |
Intersecting Lines (different slopes) |
Perpendicular Lines (product of the slopes is -1) |
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| Graph |  |
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| Functional Form (Slope-intercept Form) |y=mx+b| |
|m_1 = m_2| and |b_1 \neq b_2| |
|m_1 = m_2| and |b_1 = b_2| |
|m_1 \neq m_2| | |m_1 \times m_2 = -1| |
| General form |\small Ax+By+C=0| |
|\dfrac{-A_1}{B_1} = \dfrac{-A_2}{B_2}| and |\dfrac{-C_1}{B_1} \neq \dfrac{-C_2}{B_2}| |
|\dfrac{-A_1}{B_1} = \dfrac{-A_2}{B_2}| and |\dfrac{-C_1}{B_1} = \dfrac{-C_2}{B_2}| |
|\dfrac{-A_1}{B_1} \neq \dfrac{-A_2}{B_2}| | |\dfrac{-A_1}{B_1}\times \dfrac{-A_2}{B_2} = -1| |
It is easiest to compare straight lines when they are in functional form, or slope-intercept form |y = mx + b|.
Parallel lines never intersect in the plane, since they have the same slope.
Since two parallel lines have exactly the same slope, they never intersect.
We make a distinction between distinct parallel lines and coincident parallel lines.
Non-intersecting parallel lines (also called non-coincident or disjoint) are parallel lines separate from each other.
Graphically, two non-intersecting parallel lines are represented in the following way:
Using the equations, two lines can be identified as non-intersecting parallel lines when their slopes are the same (since they are parallel lines), but their y-intercepts are different (since the lines are separate from each other).
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The equations |y = \color{red}{4}x \color{blue}{- 2}| and |y = \color{red}{4}x \color{blue}{+ 9}| represent non-intersecting parallel lines since their slopes are the same, but their y-intercepts are different.
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Consider the following equations: |3y = 2x + 1| and |y = \color{red}{\frac{2}{3}}x \color{blue}{+ 4}|. We must transform the first equation into functional form, or slope-intercept form, to be able to compare the lines. |y = \color{red}{\frac{2}{3}}x \color{blue}{+ \frac{1}{3}}| is obtained for the first equation. We notice that the slopes of the lines are identical, but the y-intercepts are different.
Algebraically solving a system of equations with non-intersecting parallel lines results in an impossible operation meaning there is no solution.
Coincident parallel lines are identical lines which have the same equation.
Graphically, two coincident parallel lines are represented in the following way:
Using equations, two lines can be identified as coincident parallel lines when their slopes are identical (since they are parallel lines) and their y-intercepts are identical (so the lines overlap each other).
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The lines with equations |y = \color{blue}{-1} + x| and |y = x \color{blue}{- 1}| are coincident since their y-intercepts are identical and their slopes (the slopes are both equal to 1) are equal.
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Consider the following equations: |\color{blue}{4} = y + \color{red}{2}x| and |y = \color{red}{-2}x \color{blue}{+ 4}|. Transform the first equation into functional form, or slope-intercept form, to compare them, resulting in |y = \color{red}{-2}x \color{blue}{+ 4}| for the first equation. We see that the slopes and the y-intercept are identical.
Algebraically solving a system of equations of two coincident parallel lines results in an equality, meaning there are an infinite number of solutions.
Intersecting lines are lines which intersect in the plane at a single point, since they do not have the same slope.
Since two intersecting lines do not have the same slope, these lines intersect at only one point.
Graphically, two intersecting lines can be represented in the following way:
Using equations, two lines can be identified as intersecting lines when their slopes are different (since they are not parallel lines).
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The lines with equations |y = \color{red}{2}x \color{blue}{+ 3}| and |y = \color{red}{5}x \color{blue}{+ 1}| intersect, since their slopes are different.
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The lines with equations |y = x| and |y = \color{red}{10}x \color{blue}{- 5}| intersect, since their slopes are different.
Perpendicular lines are also intersecting lines since they intersect at one point on the plane and they do not have the same slope.
Perpendicular lines are intersecting lines that intersect at right angles, since the slope of one is the negative reciprocal of the slope of the other.
Two perpendicular lines have opposite and inverse slopes. The product of the slopes of two perpendicular lines is equal to -1.
Let |y=m_1x+b_1| and |y=m_2x+b_2| be two perpendicular lines. Then |m_{1}\times m_{2} = -1|.
The opposite of a real number |a| is |-a|. The sum of two opposite numbers is zero.
The reciprocal or inverse of a real number |a| is |\dfrac{1}{a}|. One number is the reciprocal of another if their product is 1.
Graphically, two perpendicular lines are represented the following way:
Using equations, two lines can be identified as perpendicular lines when both slopes are negative reciprocals of one another or when the product of the two slopes is -1.
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The lines with equations |y = \color{red}{\dfrac{1}{2}}x \color{blue}{+ 5}| and |y = \color{red}{-2}x \color{blue}{+ 3}| are perpendicular, because the product of the two slopes |\left(\dfrac{1}{2}\times -2\right)| is equal to |-1.|
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The equations |y = \color{red}{\dfrac{-3}{5}}x \color{blue}{- 2}| and |y = \color{red}{\dfrac{5}{3}}x \color{blue}{+ 1}| are perpendicular since the product of the slopes |\left(\dfrac{-3}{5}\times \dfrac{5}{3}\right)| is equal to |-1.|
To learn about the relationship between two lines in the Euclidean plane rather than in the context of analytical geometry, see the following page: The Relationships Between Two Lines.