Question

How do you solve the system of equations $ y = 7x - 10 $ and $ y = - 3 $ by substitution?

Answer 1

Hint: We are given a pair of linear equations in two variables. There are several methods to solve the system of equations having two variables. Since, we are asked to solve it by substitution method, we will apply it. In the substitution method, we express one variable in terms of the other variable in one equation and substitute that value in the next equation to get a linear equation in one variable only. We then solve the linear equation and substitute the value in one equation to get the value of the other variable.

Complete step-by-step answer:
(i)
We are given two equations:
 $
  y = 7x - 10 \\
  y = - 3 \;
  $
We are asked to solve these two equations by substitution method. In this method, we calculate the value of one variable in terms of the other variable and then substitute that value in the second equation. This will give us a linear equation with one variable which can be solved and when we obtain the value of one variable, we can put it in any of the given equations to obtain the value of another variable.
Therefore, here as we can see, in the second equation $ y = - 3 $ , we already have the value of $ y $ . So, we will just put this value in the first equation $ y = 7x - 10 $ in the place of $ y $ . So, we get:
 $ - 3 = 7x - 10 $
Which can also be written as:
 $ 7x - 10 = - 3 $
(ii)
Now, since we have got a linear equation in only one variable, we will directly solve it. Therefore, we need to find the value $ x $ to solve the equation. In order to find the value of $ x $ , we will shift everything to RHS except $ x $ .
So, adding $ 10 $ both the sides, we will get:
 $ 7x - 10 + 10 = - 3 + 10 $
On simplifying, it will become:
 $ 7x = 7 $
(iii)
Now we will divide $ 7x $ by $ 7 $ to get $ x $ on LHS and since we have to keep the equations balanced, we will divide both the sides by $ 7 $ . Therefore, we will get:
 $ \dfrac{{7x}}{7} = \dfrac{7}{7} $
On simplifying, we will get:
 $ x = 1 $
Since, we already have $ y = - 3 $ and now we got $ x = 1 $ , we have got values of both the variables and the system of equations is solved.
So, the correct answer is “ $ x = 1 $ ”.

Note: We can check whether the obtained result is correct or not by substituting the value we got of $ x $ in the equation $ y = 7x - 10 $ . If the obtained value of $ y $ comes out to be $ - 3 $ , we can say that our solution is correct. If we do not obtain the same result, then our solution is not correct and we should check our calculation again.