Question

How do you solve the system $y=2x$ and $x+3y=-14$ using substitution?

Answer 1

Hint: We are given two equations as $y=2x$ and $x+3y=-14$. We will first learn about the type of equation, then we will learn about the method that we can use to solve the problem. We are asked to use the substitution method, so we will learn the substitution method and then use it to solve the given problem. We will substitute $y$ as $2x$ and solve for $x$.

Complete step by step solution:
We are given the equations as $y=2x$ and $x+3y=-14$. We can see these equations have the highest power as 1 only. It means that they are linear equations. We are asked to find the solution and solutions refer to those values which when inserted in the equation satisfy the equation. So, we have to look for those values of $x$ and $y$ which satisfies the equations $y=2x$ and $x+3y=-14$. Now as we have the linear equation in two variables, so there are different ways to solve the problem. The various methods are as follows:
1. Substitution.
2. Elimination.
3. Cross multiplication.
4. Graphing.
We are asked to use the substitution method to solve this. In this method we substitute the value of one variable in terms of the other variable using our equation and then solve for that variable. We have $y=2x$ and $x+3y=-14$, so we will use $y=2x$ to substitute $y$ as $2x$. So, we will put $y=2x$ in the equation $x+3y=-14$. So, $x+3y=-14$ becomes,
$x+3\times \left( 2x \right)=-14$
On simplifying we get,
$x+6x=-14$
So, we have,
$7x=-14$
Dividing both sides by 7, we get,
$\begin{align}
  & \dfrac{7x}{7}=\dfrac{-14}{7} \\
 & \Rightarrow x=-2 \\
\end{align}$
We get $x$ as $-2$.
Now we use this $x=-2$ to find the value of $y$.
As we have $y=2x$, so we get,
$\begin{align}
  & y=2\left( -2 \right)\text{ }\left[ \text{as }x=-2 \right] \\
 & =-4 \\
\end{align}$
So, our solution is, $y=-4$ and $x=-2$.

Note: We can cross check our solution. We will see whether these values satisfy our equation or not. First equation that we have is $y=2x$. So, putting the values of $y=-4$ and $x=-2$, we get,
$\begin{align}
  & -4=2\times \left( -2 \right) \\
 & -4=-4 \\
\end{align}$
Which is true, so $x=-2,y=-4$ satisfies this equation. Now the second equation that we have is, $x+3y=-14$, so we put the values of $y=-4$ and $x=-2$, so we get,
$-2+3\left( -4 \right)=-14$
Simplifying, we get,
$\begin{align}
  & -2+\left( -12 \right)=-14 \\
 & -14=-14 \\
\end{align}$
So, this is also true. Hence $y=-4$ and $x=-2$ satisfy this equation too, so our solution is correct.