3 Equations with an infinite number of solutions

Some pairs of simultaneous equations can possess an infinite number of solutions. Consider the following example.

Example 31

Solve the equations

$2x+y=8$ (9)

$4x+2y=16$ (10)

Solution

If Equation (9) is multiplied by 2 we find both equations are identical: $4x+2y=16$ . This means that one of them is redundant and we need only consider the single equation

$2x+y=8$

There are infinitely many pairs of values of $x$ and $y$ which satisfy this equation. For example, if $x=0$ then $y=8$ , if $x=1$ then $y=6$ , and if $x=-3$ then $y=14$ . We could continue like this producing more and more solutions. Suppose we choose a value, say $\lambda$ , for $x$ . We can then write

$2\lambda +y=8\text{so that }y=8-2\lambda$

The solution is therefore $x=\lambda$ , $y=8-2\lambda$ for any value of $\lambda$ whatsoever . There are an infinite number of such solutions .

Exercises

Solve the given simultaneous equations by elimination:

1. 1. $5x+y=8$ , $-3x+2y=-10$ ,
   2. $2x+3y=-2$ , $5x-5y=20$ ,
   3. $7x+11y=-24$ , $-9x+y=46$
2. A straight line has equation of the form $y=ax+b$ . The line passes through the points with coordinates $\left(2,4\right)$ and $\left(-1,3\right)$ . Write down the simultaneous equations which must be satisfied by $a$ and $b$ . Solve the equations and hence find the equation of the line.
3. A quadratic function $y=a{x}^2+bx+c$ is used in signal processing to approximate a more complicated signal. If this function must pass through the points with coordinates $\left(0,0\right)$ , $\left(1,3\right)$ and $\left(5,-11\right)$ write down the simultaneous equations satisfied by $a$ , $b$ and $c$ . Solve these to find the quadratic function.

Answer