Question

How do you solve the system of equations: $2a + 2x = 18$ and $a + 3x = 17$ ?

Answer 1

Hint: In the given question, we need to solve two simultaneous equations in two variables. There are various methods to solve two given equations in two variables like substitution method, cross multiplication method, elimination method, matrix method and many more. We will solve the equations using the substitution method by substituting the value of one variable in terms of another into the second equation.

Complete step by step answer:
In the question, we are given a couple of simultaneous linear equations in two variables.
$2a + 2x = 18 - - - - \left( 1 \right)$
$\Rightarrow a + 3x = 17 - - - - \left( 2 \right)$
In the substitution method, we substitute the value of one variable from an equation into another equation so as to get an equation in only one variable.Now putting the value of a obtained from the second equation into the first one. So, we get,
$ \Rightarrow a = 17 - 3x$

Substituting the value of a in first equation, we get,
\[ \Rightarrow 2\left( {17 - 3x} \right) + 2x = 18\]
Opening the brackets, we get,
\[ \Rightarrow 34 - 6x + 2x = 18\]
Adding up like terms,
\[ \Rightarrow 34 - 4x = 18\]
Shifting the terms consisting x into right side of equation and constant terms to left side of equation, we get,
\[ \Rightarrow 34 - 18 = 4x\]
\[ \Rightarrow 16 = 4x\]

Dividing both sides by four and cancelling common factors in numerator and denominator, we get,
\[ \Rightarrow x = \dfrac{{16}}{4} = 4\]
So, the value of x is $4$.Putting the value of x in any of the two equations, we get,
$a + 3x = 17$
$ \Rightarrow a + 3\left( 4 \right) = 17$
Simplifying the expression, we get,
$ \Rightarrow a + 12 = 17$
$ \therefore a = 5$

Therefore ,the solution of the simultaneous linear equations $2a + 2x = 18$ and $a + 3x = 17$ is $x = 4$ and $a = 5$.

Note: Linear Equation in two variables: An equation consisting of 2 variables having degree one is known as Linear Equation in two variables. Standard form of Linear Equation in two variables is $ax + by + c = 0$ where a, b and c are the real numbers and a, b which are coefficients of x and y respectively are not equal to 0. Transposition method helps us to find the solution to the equation by shifting the terms from one side to another and reversing the signs.