Question

How do you solve \[7x + 2y = - 31\] and \[ - 5x + y = 27\] using substitution?

Answer 1

Hint: To solve the given simultaneous equation using substitution method, 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
 \[7x + 2y = - 31\] …………………………. 1
 \[ - 5x + y = 27\] ..………………………… 2
The standard form of simultaneous equation is
 \[Ax + By = C\]
Equation 2 can be written in terms of \[y\] to get the value of \[x\] that is
 \[ - 5x + y = 27\]
 \[ \Rightarrow \] \[y = 27 + 5x\] …………………………… 3
Hence, substitute the value of \[y\] in equation 1 as
 \[7x + 2y = - 31\]
 \[ \Rightarrow \] \[7x + 2\left( {27 + 5x} \right) = - 31\]
After substituting the y term, simplify the obtained equation
 \[7x + 54 + 10x = - 31\]
As there are common terms, let us simplify we get:
 \[17x + 54 = - 31\]
 \[17x = - 31 - 54\]
 \[17x = - 85\]
Therefore, the value of \[x\] after simplifying the terms we get
 \[ \Rightarrow \] \[x = \dfrac{{ - 85}}{{17}}\]
 \[ \Rightarrow \] \[x = - 5\]
Now we need to find the value of y, as we got the value of \[x\] , substitute the value of \[x\] as -5 in equation 3 we get,
 \[y = 27 + 5x\]
 \[y = 27 + 5\left( { - 5} \right)\]
 \[ \Rightarrow \] \[y = 27 - 25\]
 \[ \Rightarrow \] \[y = 2\]
Therefore, the value of \[y\] is 2.
Hence the values of \[x\] and \[y\] are
 \[x = - 5\] and \[y = 2\]
So, the correct answer is “ \[x = - 5\] and \[y = 2\] ”.

Note: Simultaneous equations are a set of two or more equations, each containing two or more variables whose values can simultaneously satisfy both or all the equations in the set, the number of variables being equal to or less than the number of equations in the set.
We know that Simultaneous linear 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 we need to combine all the terms and then simplify the terms by substitution method i.e., solving the equations with respect to x and y to get the value of x and the value of y.