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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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Edzai Conilias Zvobwo is passionate about empowering Africans through mathematics, problem-solving techniques and media. As such, he founded MathsGee. Through this organisation, he has helped create an ecosystem for disseminating information, training, and supporting STEM education to all African people. A maths evangelist who teaches mathematical thinking as a life skill, Edzai’s quest has seen him being named the SABC Ambassador for STEM; he has been invited to address Fortune 500 C-suite executives at the Mobile 360 North America; was nominated to represent Southern Africa at the inaugural United Nations Youth Skills Day in New York; was invited to be a contributor to the World Bank Group Youth Summit in 2016; has won the 2014 SADC Protocol on Gender and Development award for his contribution to women’s empowerment in education; and has partnered with local and global firms in STEM interventions.

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