Question

How do you solve the following system \[6x + 2y = - 4\], \[6x + 3y = - 12\]?

Answer 1

Hint: To solve the given system of equations, we will apply the elimination method.
Since the coefficient of \[x\], by subtracting these two equations we will get the value of \[y\].
Substituting the value of \[y\] in any of the equations we will get the value of \[x\].

Complete step by step answer:
It is given that; the system of equation is \[6x + 2y = - 4\], \[6x + 3y = - 12\]
We have to solve the equations.
Let us consider,\[6x + 2y = - 4\]… (1)and \[6x + 3y = - 12\]… (2)
Subtracting (1) and (2) we get,
\[ \Rightarrow (6x + 2y) - (6x + 3y) = - 4 - ( - 12)\]
Solving we get,
\[ \Rightarrow 2y - 3y = - 4 + 12\]
Solving again we get,
\[ \Rightarrow y = - 8\]
Substituting the value of \[y = - 8\] in (1) we get,
\[ \Rightarrow 6x + 2( - 8) = - 4\]
Simplifying we get,
\[ \Rightarrow 6x = 12\]
Simplifying again we get,
\[ \Rightarrow x = 2\]

Hence, the solution of the system of equation is \[x = 2\]and \[y = - 8\]

Note: Let us consider the pair of linear equations in two variables \[x\] and \[y\].
\[{a_1}x + {b_1}y + {c_1} = 0\]
\[{a_2}x + {b_2}y + {c_2} = 0\]
Here \[{a_1},{b_1},{c_1},{a_2},{b_2},{c_2}\]are all real numbers.
Case 1. If \[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}\]then there will be infinitely many solutions. This type of equation is called a dependent pair of linear equations in two variables. If we plot the graph of this equation, the lines will coincide.
Case 2. If \[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}\]then there will be no solution. This type of equation is called an inconsistent pair of linear equations. If we plot the graph, the lines will be parallel.
Mathematicians show the relationship between different factors in the form of equations. "Linear equations" mean the variable appears only once in each equation without being raised to a power.
A "system" of linear equations means that all of the equations are true at the same time. So, the person solving the system of equations is looking for the values of each variable that will make the entire equations true at the same time. If no such values can satisfy all of the equations in the system, then the equations are called "inconsistent."