Editing Elementary algebra (section)

=== Linear equations with two variables ===
[[File:Linear-equations-two-unknowns.svg|thumb|right|Solving two linear equations with a unique solution at the point that they intersect.]]
A linear equation with two variables has many (i.e. an infinite number of) solutions.<ref>Sinha, ''The Pearson Guide to Quantitative Aptitude for CAT 2/e''Publisher: Pearson Education India, 2010, {{ISBN|8131723666}}, 9788131723661, 599 pages, [https://books.google.com/books?id=eOnaFSKRSR0C&q=many+solutions&pg=PA195 page 195]</ref> For example:

:Problem in words: A father is 22 years older than his son. How old are they?
:Equivalent equation: <math>y = x + 22</math> where {{mvar|y}} is the father's age, {{mvar|x}} is the son's age.

That cannot be worked out by itself. If the son's age was made known, then there would no longer be two unknowns (variables). The problem then becomes a linear equation with just one variable, that can be solved as described above.

To solve a linear equation with two variables (unknowns), requires two related equations. For example, if it was also revealed that:
; Problem in words
: In 10 years, the father will be twice as old as his son.
;Equivalent equation
: <math>\begin{align}
y + 10 &= 2 \times (x + 10)\\
y &= 2 \times (x + 10) - 10 && \text{Subtract 10 from both sides}\\
y &= 2x + 20 - 10 && \text{Multiple out brackets}\\
y &= 2x + 10 && \text{Simplify}
\end{align}</math>

Now there are two related linear equations, each with two unknowns, which enables the production of a linear equation with just one variable, by subtracting one from the other (called the elimination method):<ref>[[Cynthia Y. Young]], ''Precalculus'', Publisher John Wiley & Sons, 2010, {{ISBN|0471756849}}, 9780471756842, 1175 pages, [https://books.google.com/books?id=9HRLAn326zEC&dq=linear+equation++two+variables++many+solutions&pg=PA699 page 699]</ref>
:<math>\begin{cases}
y = x + 22 & \text{First equation}\\
y = 2x + 10 & \text{Second equation}
\end{cases}</math>
:<math>\begin{align}
&&&\text{Subtract the first equation from}\\
(y - y) &= (2x - x) +10 - 22 && \text{the second in order to remove } y\\
0 &= x - 12 && \text{Simplify}\\
12 &= x && \text{Add 12 to both sides}\\
x &= 12 && \text{Rearrange}
\end{align}</math>

In other words, the son is aged 12, and since the father 22 years older, he must be 34. In 10 years, the son will be 22, and the father will be twice his age, 44. This problem is illustrated on the associated plot of the equations.

For other ways to solve this kind of equations, see below, '''[[#System of linear equations|System of linear equations]]'''.