Question

How do you solve the system of equations \[5x+3y=12\] and \[4x-5y=17\]?

Answer 1

Hint: From the question given we have been asked to solve \[5x+3y=12\] and \[4x-5y=17\]. We can solve the above given equations by using the process of elimination. Here elimination means we have to make any variable either x or y in both the equations the selected variable term in the both the equation should be equal and opposite signs, this should be done by multiplying the equations with the suitable number after doing this we have to add the equations the like terms will cancel and we will get the value of one variable. By using the elimination method, first we will get one variable value and by using that we have to find another variable value.

Complete step by step solution:
From the question given, it has been given that \[5x+3y=12\]
Let us assume this equation be \[\left( 1 \right)\].
\[4x-5y=17\]
Let us assume this equation be \[\left( 2 \right)\].
First of all, multiply equation \[\left( 1 \right)\] with \[4\] , we get
\[\Rightarrow 20x+12y=48\]
Let us assume this equation be \[\left( 3 \right)\]
Now, multiply equation \[\left( 2 \right)\] with\[-5\], we get
\[\Rightarrow -20x+25y=-85\]
Let it be equation \[\left( 4 \right)\]
Now, add equation \[\left( 3 \right)\] in equation \[\left( 4 \right)\].
By adding, we get
\[\begin{align}
  & \Rightarrow 20x+12y=48 \\
 & \Rightarrow -20x+25y=-85 \\
\end{align}\]
\[\Rightarrow 37y=-37\]
\[\Rightarrow y=\dfrac{-37}{37}\]
\[\Rightarrow y=-1\]
Now, substitute \[y=-1\] in the equation \[\left( 3 \right)\].
By substituting \[y=-1\] in the equation \[\left( 3 \right)\], we get
\[\Rightarrow 20x+12y=48\]
\[\Rightarrow 20x+12\times -1=48\]
\[\Rightarrow 20x-12=48\]
\[\Rightarrow 20x=48+12\]
\[\Rightarrow 20x=60\]
\[\Rightarrow x=\dfrac{60}{20}\]
\[\Rightarrow x=3\]

Therefore, the solution for the given equations is \[x=3\] and \[y=-1\].

Note: We should be very careful while doing the calculation in this problem. Also, we should know all methods to solve the given equations. we can solve this question by substitution method also. For this question we have chosen an elimination method. Like this, we have to choose their suitable method to solve the given equations.
multiply equation\[\left( 1 \right)\] with \[4\]
 multiply equation \[\left( 2 \right)\] with\[-5\], we get
In the above two steps students should know which number should be multiplied to cancel variable terms.
 Calculation should be done very carefully while finding the solution for the given question.
Student can also do substitution method.it means
\[\Rightarrow 4x-5y=17\]
\[\Rightarrow y=\dfrac{4x-17}{5}\]
Substitute this y value in another equation \[5x+3y=12\] then we will get the value of x from that we will get the value of x.
Students can also check whether in both the methods the solutions for the given equations are the same or not.