Question

How do you solve the system of equations \[2x - 5y = 10\] and \[4x - 10y = 20\]?

Answer 1

Hint: Equations that have more than one unknown can have an infinite number of solutions, finding the values within two or more equations are called simultaneous equations because the equations are solved at the same time. To solve the given simultaneous linear equation, combine all the like terms or by using any of the elementary arithmetic functions i.e., addition, subtraction, multiplication and division hence simplify the terms to get the value of \[x\] also the value of \[y\].

Complete step by step solution:
Let us write the given equation
\[2x - 5y = 10\] …………………………. 1
\[4x - 10y = 20\] ..………………………… 2
The standard form of equation is
\[Ax + By = C\]
If we re-write equation 1 i.e., \[2x - 5y = 10\]as a linear (that means in a form that can be drawn as a straight line) function in standard format, you would get:
 \[y = \dfrac{{2x - 10}}{5}\].
If we re-write equation 2 i.e., \[4x - 10y = 20\] as a function in standard form you would also get:
 \[y = \dfrac{{2x - 10}}{5}\].
Therefore, both equations represent the same line; any \[\left( {x,y} \right)\] pair that works for one also works for the other i.e., there are an infinite number of solutions for this pair of equations since they are just two different versions of the same line.

Note: If you have two linear equations (which you can think of as two straight lines drawn in the xy-plane) there are 3 possibilities: they cross at exactly one point; they don't; or they are exactly the same line so they match up at every point along the line. The key point to solve this type of equation is to combine all the like terms i.e., finding out the common term and evaluate for the variable asked, but here both equations represent the same line.