Simultaneous Equations

Tags

math

Refers to an equation based on multiple terms expressed across several constraints (equations). To solve a simultaneous equation you must find the values of the given variables that satisfy all the equations.

Solving a Simultaneous Equation

The classic way to solve a simultaneous equation is to re-arrange one of the constraints into a form equivalent to one of the terms, and then substitute it into the other equation. We repeat this until we get an equation with only a single term which we can solve. Then we substitute the found value of this term back into other equations until we resolve all the terms.

For example:

\begin{align} 3x + 2y &= 7 \label{eq:sim-eq-example-1} \
x^2 + y &= 8 \label{eq:sim-eq-example-2} \end{align}

To begin we define \(x\) in terms of \( y \) using eq:sim-eq-example-2

\begin{align*} y = 8 - x^2 \end{align*}

Then we substitute this back into eq:sim-eq-example-1 to get

\begin{align*} 3x + 2y &= 7 \
3x + 2(8 - x^2) &= 7 \
3x + 16 - 2x^2 &= 7 \
0 &= 2x^2 - 3x - 9 \end{align*}

This equation has the real value roots \[ x = \frac{{3 \pm \sqrt{9 + 4 \times 2 \times 9}}}{2 \times 2} = \frac{3 \pm \sqrt{81}}{4} = \frac{3 \pm 9}{4} \] which is

\begin{align} x =& 3 & x &= - \frac{3}{2} \end{align}

which can be substituted back into either eq:sim-eq-example-1 or eq:sim-eq-example-2 to get

\begin{align} y =& -1 & y = \frac{23}{4} \end{align}