Question

How do you solve the system $ x = 3y - 1 $ and $ x + 2y = 9? $

Answer 1

Hint: In this question, we will be solving using the substitution method that is generally used to solve the system of equations. In this method, first solve the equation for one variable, and substitution the value of the variable in the other equation. Finally we get values of $ x $ and $ y $ . Finally we get the required answer

Complete step-by-step solution:
Given,
 $ \Rightarrow x = 3y - 1 $ and
 $ \Rightarrow x + 2y = 9 $
You know the value of the variable $ x $ , so you can first substitute that into the second equation.
Let, $ x + 2y = 9 $
 $ \Rightarrow \left( {3y - 1} \right) + 2y = 9 $ . Here $ x = 3y - 1 $
Next, we remove the parentheses and solve the above equation and we get
 $ \Rightarrow 3y - 1 + 2y = 9 $
Next, we move all numbers in right hand side (RHS) and we get
 $ \Rightarrow 3y + 2y = 9 + 1 $
Now, adding the variables and numbers in both side and we get
 $ \Rightarrow 5y = 10 $
Divide by 5 in both side and we get
 $ \Rightarrow \dfrac{5}{5}y = \dfrac{{10}}{5} $
 $ \Rightarrow y = 2 $
Having said this, just replace the $ y $ in the first equation in order to get the $ x $ .
Therefore, $ x = 3(2) - 1 $
Next, we simplify the above equation.
 $ \Rightarrow x = 6 - 1 $
Let us subtract we get
 $ \Rightarrow x = 5 $

Therefore, $ (x,y) \Rightarrow (5,2) $

Note: Let us have 3 equations and 3 variables $ x,y $ and $ z $ . Now pick up an equation with $ x $ and segregate it say $ x $ in terms of $ y,z. $ when we put this value of $ x $ in two other equation we get two equations in $ y $ and $ z. $
We can now find $ y $ in terms of $ z $ say using the second equation and when we put in the third equation we get a value of $ z $ .
Once $ z $ is known, it is easy to find $ y $ and then $ x $ .
Here we can verify our answer:
 $ \Rightarrow x = 3y - 1 $
We put $ x = 5 $ and $ y = 2 $ in the above equation and we get
 $ \Rightarrow 5 = 3(2) - 1 $
Next, simplify the above equation and subtract the right hand side:
 $ \Rightarrow 5 = 6 - 1 $
 $ \Rightarrow 5 = 5 $
And for the second one:
 $ \Rightarrow x + 2y = 9 $
We put $ x = 5 $ and $ y = 2 $ in the above equation and we get
 $ \Rightarrow 5 + 2(2) = 9 $
Next, simplify the above equation and add the left hand side:
 $ \Rightarrow 5 + 4 = 9 $
 $ \Rightarrow 9 = 9 $
Both answers satisfy both equations, which makes them correct.