Mathematics for GIS Professionals

Simultaneous Equations

What is a system of simultaneous linear equations?

Firstly, ask “what is a linear equation?” It is an equation in one or more variables where each term’s degree is not more than 1. That means a variable $x$ may appear, but neither any higher power of $x$, such as $x^2$, nor any product of variables, such as $xy$, may appear. It has to be a pretty simple equation like:

$$3x + 2y – 5z = 8$$.

In fact, any linear equation can be put in the form

$$c_1x_1 + c_2x_2 + … + c_nx_n = c_0$$.

where $n$ is the number of variables, the variables are $x_1, x_2, … , x_n$ and $c_0, c_1, … , c_n$ are constants.

A system is just a collection of such linear equations, and to solve a system look for the values of the variables which make all the equations true simultaneously. For instance, if $$x$$ and $$y$$ are the variables, then an example system of linear equations is

$$5x – 2y = 4$$

$$x + 2y = 8$$

There are various ways of solving this system, and they lead to the unique solution where x = 2 and y = 3. We’ll look next at a common algorithm for solving systems of simultaneous equations called elimination.

The Elimination Method

This method for solving a pair of simultaneous linear equations reduces one equation to one that has only a single variable. Once this has been done, the solution is the same as that for when one line was vertical or parallel. This method is known as the Gaussian elimination method.

Example 2

Solve the following pair of simultaneous linear equations:

Equation 1:     $$2x + 3y = 8$$

Equation 2:     $$3x + 2y = 7$$

Step 1: Multiply each equation by a suitable number so that the two equations have the same leading coefficient. An easy choice is to multiply Equation 1 by 3, the coefficient of x in Equation 2, and multiply Equation 2 by 2, the x coefficient in Equation 1:

$$3 *$$ (Eqn 1) —>

$$3* (2x + 3y = 8)$$

—>    $$6x + 9y = 24$$

$$2 *$$ (Eqn 2)

—>   $$2 * (3x + 2y = 7)$$

—>    $$6x + 4y = 14$$

Both equations now have the same leading coefficient $= 6$

Step 2: Subtract the second equation from the first.

$$-(6x + 9y = 24$$

$$-(6x + 4y = 14)$$

$$5y = 10$$

Step 3: Solve this new equation for y.

$$y = \frac{10}{5} = 2$$

Step 4: Substitute $y = 2$ into either Equation 1 or Equation 2 above and solve for $x$. We’ll use Equation 1.

$$2x + 3(2) = 8$$

$$2x + 6 = 8$$    Subtract 6 from both sides

$$2x = 2$$           Divide both sides by 2

$$x = 1$$

Solution: $x = 1, y = 2$ or $(1,2)$.

Exercise

Solve the following simultaneous equations:

a.       $2x + 3y = 7$

b.      $x+y=2$

c.       $3x+3y=6$

d.      $x+y=10$

e.      $x-y=5$

f.        $x+4=7$

g.       $x+y=11$

h.      $2x+3y=10$

i.         $x+y=2$

j.        $8x-2y+1=0$

k.       $2x+4y=7$

l.         $x-y=4$

m.    $\frac{x}{5} + \frac{y}{2}=5$

n.  $4x + y = 9$

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