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# Find the solution for the system of equations $2x + y = 10$ and $5x - y = 18$ .

Last updated date: 29th Feb 2024
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Hint: The equation given in the question has two variables “x” and “y” that are linear equations in two variables. These are also called simultaneous equations also known as systems of equations that contain multiple variables. These equations can be solved by methods like substitution method, elimination method.

When one variable can be easily isolated in one of the two equations then this can be solved as a substitution method. The first equation should be solved for one variable. The solved equation should be substituted into the second equation.
Hence solving the first equation for “y” we have,
$2x - 2x + y = 10 - 2x \\ \Rightarrow 0 + y = 10 - 2x \\ \Rightarrow y = 10 - 2x \;$
Now substituting $10 - 2x$ for “y” in the second equation and solving for “x” we have,
$5x - \left( {10 - 2x} \right) = 18$
Expanding the brackets by multiplying the negative sign we have,
$\Rightarrow 5x - 10 + 2x = 18 \\ \Rightarrow 7x - 10 = 18 \;$
Adding $10$ both the sides in left hand side and right hand side we have,
$7x - 10 + 10 = 18 + 10 \\ \Rightarrow 7x - 0 = 28 \\ \Rightarrow 7x = 28 \;$
Dividing by $7$ ,
$\Rightarrow \dfrac{{7x}}{7} = \dfrac{{28}}{7}$
Cancelling $7$ and dividing $7$ by $28$ ,
$\Rightarrow x = 4$
Now substituting $4$ for “x” in the solution to the first equation and calculating for “y” we have,
$y = 10 - \left( {2 \times 4} \right) \\ \Rightarrow y = 10 - 8 \\ \Rightarrow y = 2 \;$
So, the correct answer is “x = 4 and y = 2”.

Note: The solution should be tested by substituting into both the equations and making sure that the equations hold true. The solution should be substituted for the second variable back into the first equation and should be solved for the first variable. Mathematically a solution is simply a pair of points that make a series of equations true. Graphically represented, a solution is a series of points where the lines of the equations intersect.