Question

How do you solve \[5x - 2y = 48 - 4x\] and \[6x + 7y = x + 6y + 33\] using substitution?

Answer 1

Hint: 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. 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
\[5x - 2y = 48 - 4x\] …………………………. 1
\[6x + 7y = x + 6y + 33\] ..………………………… 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 \[x\]i.e., for the equation;
\[6x + 7y = x + 6y + 33\]
To simplify the terms, we need to combine all the like terms of equation 2 and shift the terms as:
\[ \Rightarrow 7y - 6y = x - 6x + 33\]
\[ \Rightarrow y = - 5x + 33\]…………………………… 3
Hence, substitute the value of \[y\] from equation 3 in equation 1 as:
\[5x - 2y = 48 - 4x\]
\[ \Rightarrow 5x - 2\left( { - 5x + 33} \right) = 48 - 4x\]
After substituting the y term, simplify the obtained equation
\[ \Rightarrow 5x + 10x - 66 = 48 - 4x\]
\[ \Rightarrow 15x - 66 = 48 - 4x\]
As there are common terms, let us simplify we get:
\[15x + 4x = 48 + 66\]
\[ \Rightarrow 19x = 114\]
Therefore, the value of \[x\] after simplifying the terms we get
\[ \Rightarrow x = \dfrac{{114}}{{19}}\]
\[ \Rightarrow x = 6\]
Now we need to find the value of y, as we got the value of \[x\], substitute the value of \[x\] as 6 in equation 3 we get,
\[y = - 5x + 33\]
Substituting the value of x we get:
\[ \Rightarrow y = - 5\left( 6 \right) + 33\]
\[ \Rightarrow y = - 30 + 33\]
\[ \Rightarrow y = 3\]
Therefore, the value of \[y\] is 3.
Hence the values of \[x\] and\[y\] are
\[x = 6\] and \[y = 3\]

Note: The key point to solve the equations using substitution is that we need to solve one of the equations for one of the variables, then use that to substitute that variable out of the other equation, which you can then solve for the other variable. As 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., solve the equations with respect to x and y to get the value of x and the value of y.