Mathematics for GIS Professionals

Linear Equations

 Equations and Expressions are closely related. The primary difference between the two is an equals sign. An “equation” has a left side, a right side and an equals sign separating the sides. An “expression,” by contrast, doesn’t have any “sides” and is simply what the name suggests: An algebraic “expression.” Though sometimes it is possible to combine like terms, we are generally not expected to “do” or “solve” anything regarding expressions.

 

For example:

$$3x – 7 = 2$$

This is an EQUATION, because it has a left side, a right side, and an = sign separating the two.

$$3x – 7$$

This is an EXPRESSION, because there are no “sides” and no = sign.

Solving an equation implies finding the value of the unknown which is represented by a letter.

Linear equations

Linear equations have a degree of 1, meaning that the highest degree of the unknown is 1 e.g. $x+1=0$ is the same as saying $x^1 +1=0$, the power of $x$ determines the degree.

 

The general form of a linear equation is as follows:

$$y=mx+c$$

where $y$ is the dependent variable (vertical axis), $x$ is the independent variable (horizontal axis),  $m$ is the slope (gradient) and $c$ is the y-intercept (point when $x=0$)

The gradient is the rate of change of $y$ compared to the rate of change of $x$.

$m=\frac{y_2 – y_1}{x_2 – x_1}$

 

Solve the following linear equations:

 

 

  1.                  $x-3=17$
  2.                  $2x+5 =7$
  3.                  $ x+2 = 5 + 2x$
  4.                  $x +5x + 35 = 10 – 3x+ 4 -20x$
  5.                  $2x-14 = 5+x$
  6.                  $3(z+2) = 3(z-1)$
  7.                  $2(x-2) – (x-1) =2x-2$

 

 

Straight-Line Equations: Parallel and Perpendicular Lines

 

Parallel lines have the same slope (gradient) — and lines with the same slope are parallel.

 

Perpendicular slopes have opposite signs. The other$\mathit{ opposite}$ thing with perpendicular slopes is that their values are reciprocals; that is, you take the one slope value, and flip it upside down. Put this together with the sign change, and you get that the slope of the perpendicular line is the negative reciprocal of the slope of the original line — and two lines with slopes that are negative reciprocals of each other are perpendicular to each other. In numbers, if the one line’s slope is$ m = \frac{4}{5}$, then the perpendicular line’s slope will be $m = \frac{-5}{4}$. If the one line’s slope is $m = {–2}$, then the perpendicular line’s slope will be $m = \frac{1}{2}$.

 

Calculate the slopes and compare them.

One line passes through the points $(–1, {-2})$ and $(1, 2)$; another line passes through the points $({-2}, 0)$ and $(0, 4)$.

Are these lines parallel, perpendicular, or neither?

One line passes through the points $(0, -4)$ and $(-1, -7)$; another line passes through the points $(3, 0)$ and $(–3, 2)$. Are these lines parallel, perpendicular, or neither?

One line passes through the points $(-4, 2)$ and $(0, 3)$; another line passes through the points $(-3, –2)$ and $(3, 2)$. Are these lines parallel, perpendicular, or neither?

Determine whether $y=\frac{-1}{2x+1}$, $y=2x+1$ are parallel, perpendicular, or neither.

Tell whether the line for each pair of equations are parallel,perpendicular,or neither. $$y={-1}{2x-11}$$ and $ 16x-8y=-8 $

Line joining two points

Consider two points $AB$ with coordinates:

$A(1234.56;2345.67)$$ and $$B(1296.32;2417.38)$ then

$m_1=\frac{(y_A – y_B)}{x_A – x_B)}$

$=\frac{(2345.67 – 2417.38)}{(1234.56 – 1296.32)}$

$=\frac{(-71.71)}{(-61.76)}$

$+1.161108$

 

$c_1 = y_A -m_1x_A = 2345.67 – 1.161108 * 1234.56$

or

$c_1=y_B-m_1x_B = 2417.38 – 1.161108 * 1296.32$

 

Both cases, this gives $c_1 = 912.21$ hence the line joining $A$ to $B$ is $y=1.161108x + 912.21$

 

The point of intersection of two lines

Following on from the example in Example 3.1, for two points $CD$ with:

$C(1300.24, 2351.77)$    and   $D(1212.45, 2431.78)$ we have

$m_2 = \frac{(-80.01)}{(87.79)} = -0.9113794 $$   and   $$ c_2 = 3536.78 $

Or for the line $CD, y = -0.9113794 x + 3536.78$

The point of intersection between the lines $AB$ and $CD$ must satisfy both these conditions. Then,

For $AB: y = 1.161108 x + 912.21$

For $CD: y = -0.9113794 x + 3536.78 $

Hence, at point $P$:

$1.161108 * x + 912.21 = -0.9113794 * x + 3536.78$

Adding $0.9113794 * x $ to both sides of the equation and taking $912.21$ away from both sides gives:

$1.161108 * x + 0.9113794 * x + 912.21 – 912.21$

$= -0.9113794 * x + 0.9113794 * x + 3536.78 – 912.21$ which results in

$2.072487 * x = 2624.57$

Dividing both sides by $$2.072487$$

$x = 1266.39$

Hence, $y = 1.161108 * 1266.39 + 912.21 $ or $y = 2382.62$

The coordinates of P are therefore $(1266.39, 2382.62)$. Thus, using the principles of arithmetic, we have found the coordinates of the point of intersection of two lines.

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