Question

How do you solve the simultaneous equations $3x - 4y = 11$ and $5x + 6y = 12$

Answer 1

Hint: For solving the above question, we would be having the knowledge of solving the system of line two variables. In this question we would be using the elimination method. In this method, we will first try to make the coefficient of any one variable of the two as equal and then add or subtract the new equations accordingly and then we will get the equation on which will be having only one variable. Then we can solve the equation to get the value of that particular variable which is left and after getting the value variable, we can plug in that value in any of the equations and then get the value of the other variable

Complete step-by-step answer:
You can solve this system of equations by using the multiplication method
The first thing that you need to do is pick a variable to eliminate first, then figure out the LCM of its coefficients.
Let's say that you want to eliminate $x$ and solve for \[y\] first. The two coefficients of \[x\;\] are 3 and 5, which means that they're LCM will be equal to 15.
So, multiply the first equation by 5 and the second equation by −3 to get
\[ \Rightarrow \]\[5 \times (3x - 4y) = 5 \times 11\]
\[ \Rightarrow \]\[15x - 20y = 55\]
and the second equation by −3 to get
\[ \Rightarrow \]\[( - 3) \times (5x + 6y) = - 3 \times 12\]
\[ \Rightarrow \]\[ - 15x - 18y = - 36\]$y$
Add the left side and the right side of these two equations separately to get
\[ \Rightarrow \]\[15x - 20y - 15x - 18y = 55 - 36\]
On cancelling the above term we get only value of
$
   \Rightarrow - 38y = 19 \\
   \Rightarrow y = -\dfrac{19}{38} \\
   \Rightarrow y = -\dfrac{1}{2} \;
 $
Now use this value of y in one of the two equations to determine the value of x.
$
   \Rightarrow 3x - 4 \times -\dfrac{1}{2} = 11 \\
   \Rightarrow 3x + 2 = 11 \\
   \Rightarrow x = 11 - 2 = 3 \;
 $
The solutions to this system of equations are
\[
  x = 3 \\
  y = -\dfrac{1}{2} \;
 \]
So, the correct answer is “x=3 and $y= -\dfrac{1}{2}$”.

Note: For these equations we can also use a substitution method in which we pour value of one variable from equation one to equation 2 to get the solution of this linear equation.
Substitution Method: It is defined as the method in which value of one variable is placed in terms of another variable so as to get a single variable equation which can be solved easily simply