Question

Solve equations using substitution method.
\[3x+2y=1\] and \[2x-3y=5\]
a) 1 and – 1
b) 1 and 1
c) – 1 and – 1
d) $1\text{ and}\dfrac{1}{2}$

Answer 1

Hint: It is given we must use a substitution method. First look at the substitution method definition carefully and try to understand it. Try to convert the variable y in terms of x from any equation, by this you get an equation which has only one variable. It is called a single variable equation. Try to keep all variable terms on the left hand side and all constants on the right hand side. Algebraically find the value of a variable using the value as you know relation with other variables just substitute it to get the value of that variable. The pair of values will be your result. Juts verify them by substituting into one of the equations.

Complete step-by-step answer:
Substitution method: The method of solving a system equations. It works by solving one of the equations for one of the variables to get in terms of other variables, then plugging this back into another equation, solving for the other variable. By this you can find both the variables. This method is generally used when there are 2 variables, for more variables it will be tough to solve.
Given equations can be written in the form as given below
\[3x+2y=1\] ……. ( 1 )
\[2x-3y=5\]……( 2 )
By substituting 2y on both sides of equation (1), we get it as\[3x=1-2y\] . By dividing with 3 on both sides, we get it in form of
$x=\dfrac{1-2y}{3}............\left( 3 \right)$
By substituting this x value into equation (2) we get it as
$2\left( \dfrac{1-2y}{3} \right)-3y=5$
By taking least common multiple, we can write it as
$\dfrac{2\left( 1-2y \right)-9y}{3}=5$
By cross multiplying, we can write it in the form of,
\[2(1-2y)-9y=15\]
By multiplying 2 inside bracket to remove it, we get it as
\[2-4y-9y=15\]
By taking y as common, we can write the equation as
\[2-(4+9)y=15\]
By subtracting 2 on both sides, we get it as
\[-13y=13\]
By dividing with +3 on both sides, we get it as
\[y=-1\]
By substituting this in the equation (3) we get it as
$x=\dfrac{1-2\left( -1 \right)}{3}$
By simplifying the term we can write it in the form of
$x=\dfrac{1+2}{3}=1$
By substituting x = 1 and y = - 1 in the equation (2) we can write it as
\[2(1)-3\left( -1 \right)=5\]
By simplifying, we can write it in the form of
\[5=5\]
RHS = LHS
 The solution to the given question is \[(1,-1)\] .

So, the correct answer is “Option A”.

Note: Be careful while removing brackets, don’t forget that the constant must also be multiplied. Generally students multiply to variable and forget about constant verification of solutions 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 value of x, y won’t change.