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Module 8: Linear Equations and Linear Systems

Relations between Linear Equations

In mathematics, relations between linear equations are connections or associations between two or more linear equations.

Click on the boxes below to learn more about specific types of relations between linear equations!

Two lines are parallel when they can be placed next to each other and never meet at any point of extension from both sides. For example, if we take a horizontal line, a parallel line would be one that is also completely horizontal.

The same can be said about linear equations. A characteristic of such is that while the y-int of two different lines may be different, they are considered parallel if their slopes ( m {"version":"1.1","math":"m"}) are exactly the same.

 

Tutorial 37: How to Determine the Equation of a Line Parallel to a Given Equation that Passes through a Specific Coordinate

Example:

Find the line parallel to y = 1 2 x + 6 {"version":"1.1","math":"y=\dfrac{1}{2}x + 6"} that passes through (3,2).

 

  1. Determine the slope (or m {"version":"1.1","math":"m"}) of the new linear equation based on the slope of the given equation. As the slope of the first equation is m = 1 2 {"version":"1.1","math":"m=\dfrac{1}{2}"}, this will also be the case for the new equation.
  2. Plug the slope and coordinate of interest into the slope-intercept general form and solve for b {"version":"1.1","math":"b"}.

    y = m x + b {"version":"1.1","math":"y=mx+b"}

    2 = 1 2 ( 3 ) + b {"version":"1.1","math":"2=\dfrac{1}{2}(3)+b"}

    2 = 3 2 + b {"version":"1.1","math":"2=\dfrac{3}{2}+b"}

    2 − 3 2 = b {"version":"1.1","math":"2-\dfrac{3}{2}=b"}

    b = 1 2 {"version":"1.1","math":"b=\dfrac{1}{2}"}

    ∴ y = 1 2 x + 1 2 {"version":"1.1","math":"\therefore y=\dfrac{1}{2}x+\dfrac{1}{2}"}

  3. To visualize, graph both equations out using any of the aforementioned methods (Refer to the Graphing section in this module).

    The green line is the original, the blue is the new one. Notice that it passes through the point of interest.

Two lines are perpendicular to one another when they intersect at a 90 degree angle. Much like parallel lines, this characteristic can be reflected in linear equations. That is, two perpendicular lines will have slopes that are negative reciprocals to each other. Recall that reciprocals are the flipped version of fractions/whole numbers, the negative just needs to be added/removed afterwards.

For example, 1 2 {"version":"1.1","math":"\dfrac{1}{2}"} and − 2 {"version":"1.1","math":"-2"} are negative reciprocals to each other.

 

Tutorial 38: How to Determine the Equation of a Line Perpendicular to a Given Equation that Passes through a Specific Coordinate

Example:

Find the line perpendicular to y = 1 2 x + 6 {"version":"1.1","math":"y=\dfrac{1}{2}x + 6"} that passes through (3,2).

 

This process is almost exactly the same as that for determining a parallel line. However, remember the slope of the second line will be the negative reciprocal of the given line.

In the example, the negative reciprocal of 1 2 {"version":"1.1","math":"\dfrac{1}{2}"} is − 2 {"version":"1.1","math":"-2"}. Take this new slope and plug it into the general slope-intercept equation along with the coordinate of interest and solve for b {"version":"1.1","math":"b"}.

y = m x + b {"version":"1.1","math":"y=mx+b"}

2 = − 2 ( 3 ) + b {"version":"1.1","math":"2= -2(3)+b"}

2 = − 6 + b {"version":"1.1","math":"2= -6+b"}

2 + 6 = b {"version":"1.1","math":"2+6=b"}

b = 8 {"version":"1.1","math":"b=8"}

∴ y = − 2 x + 8 {"version":"1.1","math":"\therefore y= -2x+8"}

We can graph both out to visualize how they are perpendicular to each other.

Here, the green line is the original, and the blue is the new line that passes through (3,2).

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