Question

How do you solve \[5x - y = - 9\] and \[y + 2x = 2\] ?

Answer 1

Hint: To solve the given simultaneous 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
\[5x - y = - 9\] …………………………. 1
\[y + 2x = 2\] ..………………………… 2
The standard form of equation is
\[Ax + By = C\]
Hence, rearrange the terms of equation 2 to the standard form
\[y + 2x = 2\]
\[ \Rightarrow \] \[2x - y = 2\]…………………….. 3
Now multiply equation 3 by -1
\[ - 1\left( {2x - y} \right) = 2\left( { - 1} \right)\]
After simplifying the terms, we get
\[ - 2x + y = - 2\] ……………………. 4
Add equation 1 and equation 4 i.e.,
\[5x - y = - 9\]
\[ - 2x + y = - 2\]
Therefore, simplifying the terms we get
\[3x = - 11\]
To obtain the value of \[x\], divide both sides of the equation by 3 as
\[\dfrac{{3x}}{3} = \dfrac{{ - 11}}{3}\]
Hence, the value of \[x\] is
\[x = - \dfrac{{11}}{3}\] or \[x = - 3.67\]
Now substitute the value of \[x\] in equation 2 as
\[y + 2x = 2\]
Substitute the value of \[x\] as \[x = - \dfrac{{11}}{3}\]
\[y + 2\left( { - \dfrac{{11}}{3}} \right) = 2\]
Simplify the terms
\[y - \dfrac{{22}}{3} = 2\]
Multiply both sides of the equation by 3 i.e.,
\[y\left( 3 \right) - \dfrac{{22}}{3}\left( 3 \right) = 2\left( 3 \right)\]
\[3y - 22 = 6\]
Now add 22 on both sides of the equation to obtain the value of \[y\]
\[3y - 22 + 22 = 6 + 22\]
\[ \Rightarrow \]\[3y = 28\]
As we can see that -22 and +22 implies zero.
To get the value of \[y\]divide both sides of the equation by the same term i.e., 3
\[3y = 28\]
\[ \Rightarrow \]\[\dfrac{{3y}}{3} = \dfrac{{28}}{3}\]
\[y = \dfrac{{28}}{3} = 9.33\]
Therefore, after simplifying the terms we get the value of \[y\] as:
\[y = \dfrac{{28}}{3}\]

Note: We know that Simultaneous equations are two equations, each with the same two unknowns and are "simultaneous" because they are solved together, hence the key point to solve this kind of equations is we need to combine all the terms and then simplify the terms to get the value of $x$ and also the value of $y$.