Question

How do you solve the system: $x=y-7$ and $x+8y=2$ using substitution?

Answer 1

Hint: Problems on solving linear systems can be easily done using the substitution method. First, we must take one equation and express the equation as a function of $x$ , as a value of $y$ . Then we substitute the expression of $y$in the other equation and find the value of $x$ which is used again to find the value of $y$ .

Complete step by step solution:
The two linear equations we have are
$x+8y=2.....\left( 1 \right)$ and
$x=y-7.....\left( 2 \right)$
We take equation $\left( 1 \right)$ and subtract $8y$ from both the sides of the equation as shown below
$\Rightarrow x=2-8y$
In the above equation we see that $x$ is written as a function of $y$ .
We now take the above equation and substitute the expression of the right-hand side of the above equation in place of $x$ in equation $\left( 2 \right)$ as shown below
$\Rightarrow x=y-7$
\[\Rightarrow \left( 2-8y \right)=y-7\]
The above equation can also be written as
\[\Rightarrow y-7=2-8y\]
We further add $8y$ to both the sides of the above equation as shown below
$\Rightarrow y-7+8y=2-8y+8y$
Further simplifying the above equation, we get
$\Rightarrow 9y-7=2$
Also, we add $7$ to both the sides of the above equation and get
$\Rightarrow 9y=2+7$
Further simplifying the above equation, we get
$\Rightarrow 9y=9$
Dividing the above equation by $9$ we get
$\Rightarrow \dfrac{9y}{9}=\dfrac{9}{9}$
Further simplifying the above equation, we get
$\Rightarrow y=1$
Now, we take the value of $y$ and put it in any of the given equations to get the value of $x$ .
We put the value of $x$ in the equation $\left( 2 \right)$ as shown below
$\Rightarrow x=\left( 1 \right)-7$
We simplify the above equation and get
$\Rightarrow x=-6$

Therefore, the solution of the given linear system is $x=-6$ and $y=1$ .

Note: While using the substitution method we have to keep in mind that must properly substitute the expression in the equation so that mistakes can be avoided. Also, we must be careful about the signs of the terms, as students often get confused between the positive and negative signs.