Question

How do you solve$x+2y=3$ and $3x-y=-5$?

Answer 1

From the question given, we have been asked to solve $x+2y=3$ and $3x-y=-5$ . We can solve the equations by using substitutions and transformations. We will solve the second equation and get the value of $y$ in terms of $x$ and substitute it in the first equation then we will solve the 2 equations.

Complete answer:
Given equations are $x+2y=3$
Let this equation be $\left( 1 \right)$
Another equation given is $3x-y=-5$
Let this equation be$\left( 2 \right)$
Now, what we have to do is solve for one variable in terms of another.
So, now solve the equation $\left( 2 \right)$ for the variable $y$
Equation $\left( 2 \right)$ is $3x-y=-5$ , shift $-5$ to the left hand side of the equation. By shifting $-5$ to the left hand side of the equation, we get $3x-y+5=0$
Now, shift $-y$ to the right hand side of the equation. By shifting $-y$ to the right hand side of the equation, we get $3x+5=y$
By rearranging the equation, we get $y=3x+5$
Let this equation be $\left( 3 \right)$
Now, substitute the equation $\left( 3 \right)$ in the equation $\left( 1 \right)$ to get the variable $x$
By substituting the equation $\left( 3 \right)$ in the equation $\left( 1 \right)$, we get
$x+2y=3$
$\Rightarrow x+2\left( 3x+5 \right)=3$
On furthermore simplifying the equation, we get
$x+6x+10=3$
$\Rightarrow 7x+10=3$
Now, shift $10$ to the right hand side of the equation, by shifting $10$ to the right hand side of the equation, we get
$7x=3-10$
$\Rightarrow 7x=-7$
Now, shift $7$ to the right hand side of the equation. By shifting $7$ to the right hand side of the equation, we get
$x=\frac{-7}{7}$
$\Rightarrow x=-1$
Now, substitute $x=-1$ in equation $\left( 3 \right)$. By substituting $x=-1$ in the equation $\left( 3 \right)$, we get
$y=3x+5$
$\Rightarrow y=3\left( -1 \right)+5$
$\Rightarrow y=-3+5$
$\Rightarrow y=2$
Therefore, $x=-1\text{ and }y=2$
Hence, the given question is solved.

Note: If we have any doubts in the values they got, they can verify the values by substituting the values in the given equations. If the values you got satisfy the given equations, then the values you got are correct values. We should be very careful while doing the calculation process. We should be very careful while doing the substitutions. . Similarly any set of linear equations can be solved.