Question #1553a

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Answer 1 · 2017-06-06T19:28:59

Eliminating one variable allows you to solve for the other.

You cannot solve an equation that has $2$ $2$ variables because there are infinitely many solutions.

As soon as you have 2 equations, there is a specific solution.

However, you need to get rid of one of the variables temporarily so that you can find a value for the one that is left.

This is done by making sure that the variable that is to be eliminated has the same numerical coefficients.

Case 1: If they have the SAME sign, subtract the two equations:

Consider the equations below:

$5 x + 5 y = 30$ $5xcolor(blue)(+5y) = 30$
$3 x + 5 y = 22$ $3x color(blue)(+5y) =22$

Subtracting the two equations will lead to

$2 x + 0 y = 8 \leftarrow$ $2x color(blue)(+0y) = 8" "larr$ the $y$ $y$ -terms have been eliminated

The difference between the two equations represents the difference in just the $x$ $x$ terms.

Case 2: If they have different signs, ADD the two equations:

Consider the equations below: the $x$ $x$ -terms are additive inverses

$+ 4 x + 3 y = 29$ $color(red)(+4x)+3y = 29$
$- 4 x + 5 y = - 5$ $color(red)(-4x) +5y =-5$

Adding the two equations leads to:

$0 x + 8 y = 24 \leftarrow$ $color(red)(0x)+8y=24" "larr$ the $x$ $x$ -terms have been eliminated

The sum of the equations gives us the sum of just the $y$ $y$ -terms