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# How do you solve the following linear system: $y=-3x+4$ and $y-3x=-2$ ?

Last updated date: 13th Sep 2024
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Hint: We can do this type of problems on solving linear systems easily using the substitution method. First, we take one equation and rewrite the equation as a function of $x$ and express it as a value of $y$ . Then we substitute the expression of $y$in the other equation and find the value of $x$ which we use again to get the value of $y$ .

The two linear equations we have are
$y-3x=-2.....\left( 1 \right)$ and
$y=-3x+4.....\left( 2 \right)$
We take equation $\left( 1 \right)$ and add $3x$ to both the sides of the equation as shown below
$y=-2+3x$
In the above equation we see that $y$ is written as a function of $x$ .
We now take the above equation and substitute the expression of the right-hand side of the above equation in place of $y$ in equation $\left( 2 \right)$ as shown below
$\Rightarrow y=-3x+4$
$\Rightarrow \left( -2+3x \right)=-3x+4$
We further add $3x$ to both the sides of the above equation as shown below
$\Rightarrow -2+3x+3x=-3x+4+3x$
Further simplifying the above equation, we get
$\Rightarrow -2+6x=4$
Also, we add $2$ to both the sides of the above equation and get
$\Rightarrow -2+6x+2=4+2$
Further simplifying the above equation, we get
$\Rightarrow 6x=6$
Dividing the above equation by $6$ we get
$\Rightarrow \dfrac{6x}{6}=\dfrac{6}{6}$
Further simplifying the above equation, we get
$\Rightarrow x=1$
Now, we take the value of $x$ and put it in any of the given equations to get the value of $y$ .
We put the value of $x$ in the equation $\left( 2 \right)$ as shown below
$\Rightarrow y=-3\left( 1 \right)+4$
Further, multiplying $3$ with $1$ we get
$\Rightarrow y=-3+4$
We simplify the above equation and get
$\Rightarrow y=1$

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

Note: Problems on solving linear systems can also be done using other methods. Methods such as graphical method, algebraic method or elimination method can be used to solve linear systems. Also, while using the substitution method we must be very careful about properly substituting the expression in the equation so that mistakes are avoided.