Question

How do you solve the simultaneous equation \[5c + 4d = 23\] and \[3c - 5d = - 1\]?

Answer 1

Hint: In this question, we have to solve the given algebraic equations.
First we need to express one variable of the first equation in terms of the other variable. Then putting this in the second equation we will get the value of one variable. After that, putting the value of that variable in the first equation we will get the value of the other variable and the required solution.

Complete step-by-step solution:
It is given that, \[5c + 4d = 23\] and \[3c - 5d = - 1\].
We need to solve the two equations.
Now,\[5c + 4d = 23 \ldots \ldots \ldots (i)\]
\[3c - 5d = - 1 \ldots .(ii)\]
Multiply equation i) by \[3\] we get,
\[3 \times 5c + 3 \times 4d = 3 \times 23\]
Or,\[15c + 12d = 69 \ldots ..(iii)\]
Multiply equation ii) by \[5\] we get,
\[5 \times 3c - 5 \times 5d = 5 \times \left( { - 1} \right)\]
Or,\[15c - 25d = - 5 \ldots ..(iv)\]
Subtracting equation iv) from equation iii) we get,
\[15c + 12d - \left( {15c - 25d} \right) = 69 - \left( { - 5} \right)\]
Simplifying we get,
\[15c + 12d - 15c + 25d = 69 + 5\]
On cancel the same term and adding the remaining we get
Or,\[37d = 74\]
Let us divide 37 on both sides we get,
Or,\[d = \dfrac{{74}}{{37}} = 2\]
Putting the value of d in equation i) we get,
\[5c + 4 \times 2 = 23\]
On multiplication we get
\[5c = 23 - 8\]
Simplifying we get,\[c = \dfrac{{15}}{5} = 3\]
Therefore we get,
\[c = 3,d = 2\]

Hence the solution of the simultaneous equation \[5c + 4d = 23\] and \[3c - 5d = - 1\] is \[c = 3,d = 2\].

Note: We can solve a system of linear equations by the following methods
Elimination method
Substitution method
Elimination method:
In the elimination method, the object is to make the coefficient of one variable the same in both equations so that one variable can be eliminated either by adding the equations together or by subtracting one from the other.
Substitution method:
In the substitution method, one equation is manipulated to express one variable in terms of the other. Then the expression is substituted in the other equation.
Two equations with the same variables are called systems of equations and the equations in the system are called simultaneous equations. To solve a system of equations means to find an ordered pair of numbers that satisfies both the equations in the system.