Question

How do you solve the system of equations \[2p + 3q = 1\] and \[4p - 5q = 13\] ?

Answer 1

Hint: In order to determine the value of the two systems of equations. First, we need to expand the equation \[2p + 3q = 1\] and \[4p - 5q = 13\] . From the equation we get \[p\] and \[q\] values and by substituting the value into the equation. In mathematics, a series of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are found by using substitution methods. \[\]

Complete step-by-step answer:
In the given problem, we have the two systems of equations are different.
Let the first equation, \[2p + 3q = 1 \to (1)\]
Let the second equation, \[4p - 5q = 13 \to (2)\]
We use a substitution method to solve the above equation. So, we have to expand the equation to get the value of \[p\] .
 \[(1) \Rightarrow 2p = 1 - 3q\]
Here, we need to perform division on both sides by \[2\] to get the \[p\] value. Now, we get
 \[p = \dfrac{{1 - 3q}}{2}\] .
We use a substitution method here.
By substitute the value into the equation \[(2)\]
 \[(2) \Rightarrow 4\left( {\dfrac{{1 - 3q}}{2}} \right) - 5q = 13\]
By simplify the fraction values, we can get
 \[
  (2) \Rightarrow 2(1 - 3q) - 5q = 13 \\
  2 - 6q - 5q = 13 \\
 \]
We expand the equation on RHS, we get
 \[
   - 11q = 13 - 2 \\
  q = - \dfrac{{11}}{{11}} \;
 \]
Therefore, the value of \[q = - 1\]
Now, we are going to substitute it on the equation \[(1)\] .
 \[(1) \Rightarrow 2p + 3( - 1) = 1\]
By simplify in further step by expanding the equation on RHS, we can get
 \[
  (1) \Rightarrow 2p - 3 = 1 \\
  2p = 1 + 3 \\
  p = \dfrac{4}{2} \;
 \]
Therefore the value of \[p = 2\] .
As a result, we have the common solution of the equation \[2p + 3q = 1\] and \[4p - 5q = 13\] is \[p = 2,q = -1\] .
So, the correct answer is “ \[p = 2,q = -1\] ”.

Note: We need to understand the question carefully and remember here we use a simple substitution method for finding the common solution.
Try to solve the equation using a substitution method.
First we get the value of \[p\] then we have to substitute on any equation afterward we get another solution \[q\] .
Let us check the answer and always keep the final answer simplified.