Simultaneous equations

Linear programming is a branch of mathematics which deals with either minimizing the cost or maximizing the profit.

• It gives the best way of utilizing the scarce resources available.
• It is so called because it only involves equations and inequalities which are linear.

One of the methods used in solving linear simultaneous equations is a graphical method. Two linear simultaneous equations in two unknowns can be graphically solved by passing through the following procedures.

1. Draw the two lines which represent the two equations on the xy–plane this is done by determining at least two points through which each line passes, the intercepts are commonly used.
2. Determine the point of intersection of the two lines. This point of intersection is the solution to the system of equations.

• If two straight lines are not parallel then they meet at only one point.
• In case the lines do not meet, there is no solution to the corresponding system of simultaneous equations.

Graphically solve the following system of simultaneous equations.

Solution

Determine where the lines cut the coordinate axes.

For $2x - y = 1$, the intercepts are $(\frac{1}{2}, 0)$ and $(0, -1)$.

For $3x - 3y = 6$, divide by 3 to get $x - y = 2$.

Intercepts are $(2, 0)$ and $(0, -2)$.

From the graph you can observe that the two lines meet at the point (1,1) and thus $(x,y) = (1,1)$ or $x = 1$ and $y = 1$ is the solution to the system of equations.

Find the solution to the following system of simultaneous equations by graphical method.

Solution

The line $3x + y = 9$ passes through the points $(0, 9)$ and $(3, 0)$, while the line $2x - y = 1$ passes through the points $(0, -1)$ and $\left(\frac{1}{2}, 0\right)$.

From the graph above, the two lines meet at (2, 3), therefore the values of x and y that satisfy the system of equations are 2 and 3 respectively, that is $x = 2$ and $y = 3$.

Note that you can check the obtained solution by substituting the values of $x$ and $y$ in the equations or solve the system of equations by elimination/substitution method.

Solving the system of equations in example 2 by elimination method gives the same values of $x$ and $y$ obtained by graphical method:

$\begin{aligned} 1. \quad &\begin{cases} 3x + y = 9 \\ 2x - y = 1 \end{cases} \\ &\quad 5x + 0y = 10 \\ &\quad 5x = 10, \text{this implies } x = 2 \\ &\quad \text{And } 2x - y = 1 \text{ implies } 2 \times 2 - y = 1 \\ &\quad \text{Or } 4 - y = 1 \\ &\quad \text{So } -y = 1 - 4 \\ &\quad -y = -3, \text{dividing by -1 each side gives } y = 3 \\ &\quad \text{Therefore } (x, y) = (2, 3) \end{aligned}$

Solve the following simultaneous equations graphically and check your solution by a non-graphical method:

Solution

Rearranging the equation (i), gives $2x - 3y = -3$ and $3x - 2y = 3$.

So the line $2x - 3y = -3$ passes through $(0,1)$ and $(-\frac{3}{2}, 0)$ while the line $3x - 2y = 3$ goes through the points $(0, -\frac{3}{2})$ and $(1, 0)$.

From the graph above, the lines meet at the point (3, 3), so $x = 3$ and $y = 3$.

By substitution method

From equation (ii):

Substituting (iii) into (i):

Now substitute $y = 3$ into equation (iii):

So, $x = 3$ and $y = 3$, which is the same solution obtained using the graphical method.