Question

Find the value of \[x\] and \[y\] using the cross multiplication method:
\[x + y = 15\] and \[x - y = 3\]
A) \[\left[ {9, - 6} \right]\]
B) \[\left[ { - 9,6} \right]\]
C) \[\left[ {9,6} \right]\]
D) \[\left[ { - 9, - 6} \right]\]

Answer 1

Hint:
We are required to find the values of the unknown variables using the cross-multiplication method. We will first convert the equations into the standard form \[ax + by + c = 0\]. We will then find the value of the different parameters required in the formula. Then, we will substitute all the values in our formula and equate the variable part to the constant part to get the answer.

Formula Used: We will use the formula \[\dfrac{x}{{{b_1}{c_2} - {b_2}{c_1}}} = \dfrac{y}{{{a_1}{c_2} - {a_2}{c_1}}} = \dfrac{1}{{{a_1}{b_2} - {a_2}{b_1}}}\] where \[{a_1}\] and \[{b_1}\] are the coefficients of \[x\], \[y\] and \[{c_1}\] is the constant in the first equation, respectively and \[{a_2}\] and \[{b_2}\] are the coefficients of \[x\], \[y\] and \[{c_2}\] is the constant in the second equation, respectively.

Complete step by step solution:
We have to find the value of the unknown variables \[x\] and \[y\] using the substitution method. To do that let us first write our equations in the form of \[{a_1}x + {b_1}y + {c_1} = 0\] and \[{a_2}x + {b_2}y + {c_2} = 0\]. Thus, the equations will become,
\[\begin{array}{l}x + y - 15 = 0 \to \left[ 1 \right]\\x - y - 3 = 0 \to \left[ 2 \right]\end{array}\]
Form the above equations we can see that,
\[\begin{array}{l}{a_1} = 1\\{b_1} = 1\\{c_1} = - 15\\{a_2} = 1\end{array}\]
\[\begin{array}{l}{b_2} = - 1\\{c_2} = - 3\end{array}\]
Substituting these values in the formula for the cross-multiplication, we get,
\[\dfrac{x}{{1\left[ { - 3} \right] - \left[ { - 1} \right]\left[ { - 15} \right]}} = \dfrac{y}{{\left[ { - 15} \right]1 - \left[ { - 3} \right]1}} = \dfrac{1}{{1\left[ { - 1} \right] - 1\left[ 1 \right]}}\]………..\[\left[ 3 \right]\]
We will simplify equation \[\left[ 3 \right]\] to get the value of \[x\] and \[y\] as follows:
First, we will find the value of \[x\] using equation \[\left[ 3 \right]\].
\[\dfrac{x}{{1\left[ { - 3} \right] - \left[ { - 1} \right]\left[ { - 15} \right]}} = \dfrac{1}{{1\left[ { - 1} \right] - 1\left[ 1 \right]}}\]
By cross multiplication, we get
\[ \Rightarrow x = \dfrac{{1\left[ { - 3} \right] - \left[ { - 1} \right]\left[ { - 15} \right]}}{{1\left[ { - 1} \right] - 1\left[ 1 \right]}}\]
Simplifying the above equation, we get
\[ \Rightarrow x = \dfrac{{ - 3 - 15}}{{ - 1 - 1}}\]
Further simplifying the equation, we get
\[\begin{array}{l} \Rightarrow x = \dfrac{{ - 18}}{{ - 2}}\\ \Rightarrow x = 9\end{array}\]
Now we will find the value of \[y\] using the equation \[\left[ 3 \right]\].
\[\dfrac{y}{{\left[ { - 15} \right]1 - \left[ { - 3} \right]1}} = \dfrac{1}{{1\left[ { - 1} \right] - 1\left[ 1 \right]}}\]
By cross multiplication, we get
\[ \Rightarrow y = \dfrac{{\left[ { - 15} \right]1 - \left[ { - 3} \right]1}}{{1\left[ { - 1} \right] - 1\left[ 1 \right]}}\]
Simplifying the above equation, we get
\[ \Rightarrow y = \dfrac{{ - 15 + 3}}{{ - 1 - 1}}\]
Further simplifying the equation, we get
\[\begin{array}{l} \Rightarrow y = \dfrac{{ - 12}}{{ - 2}}\\ \Rightarrow y = 6\end{array}\]
 The value of \[x\] is 9, and the value of \[y\] is 6.

Since the answers match with option [C], hence, it is the correct option.

Note:
The alternate method to solve the question can be as follows –
We will use the method of substitution to solve the equations.
\[x + y = 15\]………..\[\left[ 1 \right]\]
\[x - y = 3\]………….\[\left[ 2 \right]\]
We will find the value of the variable
\[x\] in terms of the variable \[y\] using equation
\[\left[ 1 \right]\].
\[x + y = 15\]
\[ \Rightarrow x = 15 - y\]……………\[\left[ 3 \right]\]
Now we will substitute the value of \[x\]obtained from equation \[\left[ 3 \right]\] in equation \[\left[ 2 \right]\].
\[x - y = 3\]
\[ \Rightarrow 15 - y - y = 3\]…………\[\left[ 4 \right]\]
We will now solve the equation \[\left[ 4 \right]\] further to obtain the value of the unknown variable \[y\]. So, to find \[y\] we will combine all the like terms and add them.
\[\begin{array}{l} \Rightarrow - y - y = 3 - 15\\ \Rightarrow - 2y = - 12\end{array}\]
Dividing both side by \[ - 2\], we get
\[\begin{array}{l} \Rightarrow y = \dfrac{{ - 12}}{{ - 2}}\\ \Rightarrow y = 6\end{array}\]
Now, we will substitute the value of \[y\] obtained to equation \[\left[ 1 \right]\], to get the value of the unknown variable
\[x\].
\[\begin{array}{l}x + y = 15\\ \Rightarrow x + 6 = 15\\ \Rightarrow x = 15 - 6\\ \Rightarrow x = 9\end{array}\]
The value of \[y\] is 6, and the value of \[x\] is 9.