Question

Solve the following system of equations by using the method of substitution,
$3x-5y=-1$ and $x-y=-1$.

Answer 1

Hint: It is given we must use a method of substitution. So take the second equation and send the x on the right side. So, by this you have a value of y. Now substitute the value of y in terms of x into the first equation, by this you get an equation only in terms of x. By sending all constants on the right hand side you can get x value. By using the x value, try to find the value of y. This pair of x, y values are the required result.

Complete step-by-step solution -
Substitution Method: The method of solving a system of equations. It works by solving one of the equations for one of the variables to get in terms of another variable, then plugging this back into another equation, and solving for the other variable. By this you can find both the variables. Then method is generally used when there are 2 variables. For more variables it will be tough to solve.
Given equations which we need to solve are given by:
$3x-5y=-1$ …………………..(1)
$x-y=-1$ ……………………(2)
By dividing with $-1$ on both sides of equation (2), we get:
$\Rightarrow y-x=1$
By adding with x on both sides of above equation, we get:
$\Rightarrow y-x+x=1+x$
By cancelling common term on left side of equation, we get:
$y=x+1$ ……………………………(3)
By substituting equation (3) into the equation (1), we get:
$\Rightarrow 3x-5\left( x+1 \right)=-1$
By multiply constant $-5$ into the terms inside bracket to remove it, we get:
$\Rightarrow 3x-5x-5=-1$

By adding 5 on both sides of equation, we get:
$\Rightarrow 3x-5x=-1+5$
By taking x common on left hand side of equation, we get:
$\Rightarrow x\left( 3-5 \right)=4$
By dividing with $-2$ on both sides, we get it as:
$\Rightarrow \dfrac{\left( -2 \right)x}{\left( -2 \right)}=\dfrac{4}{\left( -2 \right)}$
By simplifying the above equation, we get value of x as:
$x=-2$
By substituting this x into equation (3), we get value of y as:
$\Rightarrow y=1-2=-1$
For verification, we substitute these x, y values into the equation (1)
By substituting $x=-2,y=-1$ into equation (1), we get:
$3\left( -2 \right)-5\left( -1 \right)=-1$
We multiply the constants to remove brackets.
By doing as above, we turn the above equation to:
$\Rightarrow -6+5=-1$
By simplifying the above left hand side, we get it as:
$\Rightarrow -1=-1$
So, LHS $=$ RHS.
Hence, verified.
Therefore, the solution of given equations is $\left( -2,-1 \right)$ .

Note: Be careful while removing brackets. Don’t forget that the constant must also be multiplied. Generally, students multiply variables and forget about constant. Verification of a solution must be done to prove that our result is correct. Similarly, you can first find x in terms of y and then substitute and continue. Anyways you will get the same result because the values of x, y won’t change.