Question

How do you solve \[4x - 3y = 17\] and \[5x + 4y = 60\] ?

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
\[4x - 3y = 17\] …………………………. 1
\[5x + 4y = 60\] ..………………………… 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
\[5x + 4y = 60\]
\[ \Rightarrow \] \[4y = 60 - 5x\]
Simplifying the equation, we get
\[y = - \dfrac{{5x}}{4} + \dfrac{{60}}{4}\]
Hence, we get
\[y = - \dfrac{{5x}}{4} + 15\] …………………………… 3
Hence, substitute the value of \[y\] in equation 1 as
\[4x - 3y = 17\]
\[ \Rightarrow \]\[4x - 3\left( { - \dfrac{{5x}}{4} + 15} \right) = 17\]
After substituting the y term, simplify the obtained equation
\[\dfrac{{31x}}{4} = 62\]
Which implies that
\[31x = 62\left( 4 \right)\]
\[31x = 248\]
Therefore, the value of \[x\] after simplifying the terms we get
\[x = \dfrac{{248}}{{31}}\]
\[x = 8\]
Equation 3 is
\[y = - \dfrac{{5x}}{4} + 15\]
As we got the value of \[x\], substitute the value of \[x\] as 8 in equation 3 we get,
\[y = - \dfrac{{5x}}{4} + 15\]
\[y = - \dfrac{5}{4}\left( 8 \right) + 15\]
\[y = 5\]
Therefore, the value of \[y\] is 5.
Hence the values of \[x\] and \[y\] are
\[x = 8\] and \[y = 5\]

Additional Information:
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.

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 these kinds of equations we need to combine all the terms and then simplify the terms to get the value of \[x\] also the value of \[y\].