Question

Solve the equations using substitution method: $2x + 3y = 13$ and $4x + 5y = 23$
A. $( - 2,3)$
B. $(2,3)$
C. $(2, - 3)$
D. $( - 2, - 3)$

Answer 1

Hint: First, if the given equations are two equations and also it is linear, we can solve these in various ways like elimination method, substitution method, and matrix method. We will solve the given equations in the substitution method that is what we are also asked to do. In this method we will first try to get an expression for any one of the unknown variables and then we will substitute it in another equation to make it a linear equation with only one unknown variable. Then we will simplify it to find that unknown variable. After finding it we will substitute it in any one of the given equations to find another variable’s value.

Complete step by step solution:
Let’s take $2x + 3y = 13$ -- (1)
and
$4x + 5y = 23$ --- (2)
Let us convert the one equation into any variable form like take the equation (2)
$4x + 5y = 23 $
$\Rightarrow 4x = 23 - 5y $
$\Rightarrow x = \dfrac{{23 - 5y}}{4}$
Now applying the values in equation one we get,
$2x + 3y = 13 $
$\Rightarrow 2(\dfrac{{23 - 5y}}{4}) + 3y = 13$
Further solving we get, cancel the common terms,
$\dfrac{{23 - 5y}}{2} + 3y = 13$
By the cross multiplication we have
$23 - 5y + 6y = 26$
Hence, we get the value as
$23 - 5y + 6y = 26$
$\Rightarrow y = 26 - 23 = 3$
which is $y = 3$
Now substitute the value in equation one, we get
$2x + 3y = 13 $
$\Rightarrow 2x + 3(3) = 13 $
$\Rightarrow 2x = 4$
Hence we get $x = 2$
Therefore, they got the values as $x = 2,y = 3$ and So, option (B) is correct.

Note: As we have mentioned in the hint section, we can also solve this problem by another method called elimination method where we will rewrite the given equations by multiplying or dividing it with any number so that we will get the same term in both equations. Then we will subtract those two equations to eliminate that same term from both the equation then we will get an equation with only one variable by simplifying it we will get the value of that variable then we will substitute it in any one of the give equations to find the value of another variable.