Mathguru – Help – To solve the system of equations using cross multiplication method. - Pair of Linear Equations in Two Variables

Solved Example > No. SE 7

To solve the system of equations using cross multiplication method.

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Explanation:

In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables. For example,

$\begin{alignat}{7} 3x &&\; + \;&& 2y &&\; - \;&& z &&\; = \;&& 1 & \\ 2x &&\; - \;&& 2y &&\; + \;&& 4z &&\; = \;&& -2 & \\ -x &&\; + \;&& \tfrac{1}{2} y &&\; - \;&& z &&\; = \;&& 0 & \end{alignat}$

is a system of three equations in the three variables $x$ , $y$ , $z$ . A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by

$\begin{alignat}{2} x & = & 1 \\ y & = & -2 \\ z & = & -2 \end{alignat}$

since it makes all three equations valid.

http://en.wikipedia.org/wiki/System_of_linear_equations

In elementary arithmetic, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable.

For an equation like the following:

$\frac a b = \frac c d$ (Note that "b" and "d" must be non-zero for these to be real fractions)

one can cross-multiply to get

$ad = bc \quad \mathrm {or} \quad a = \frac {bc} {d}.$

Procedure (Our solved example in mathguru.com uses this concept).

In practice, the method of cross-multiplying means that we multiply the numerator of each (or one) side by the denominator of the other side, effectively "crossing" the terms over.

$\frac a b \nwarrow \frac c d \quad \frac a b \nearrow \frac c d.$

The mathematical justification for the method is from the following mathematical procedure.

If we start with the basic equation:

$\frac {a} {b} = \frac {c} {d}$

We can multiply the terms on each side by the same number and the terms will remain equal. Therefore, if we multiply the fraction on each side by the product of the denominators of both sides - $(bd)\,\!$ - we get:

$\frac {a} {b} \times {bd} = \frac {c} {d} \times {bd}$

We can reduce the fractions to lowest terms by noting that the b's on the left hand side and the d's on the right hand side cancel, leaving:

$ad = bc \,$ .

and we can divide both sides of the equation by any of the elements - in this case we will use "d" - yielding:

$a = \frac {bc} {d}.$

http://en.wikipedia.org/wiki/Cross-multiplication

The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.