Question

Use substitution method to solve the system of equations below. $\begin{array}{c}\mathbf{y}\mathbf{=}\mathbf{4}\mathbf{x}\mathbf{-}\mathbf{7}\\ \mathbf{3}\mathbf{x}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{-}\mathbf{1}\end{array}$

F $\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{5}\mathbf{)}$ G $\mathbf{(}\mathbf{4}\mathbf{,}\mathbf{-}\mathbf{1}\mathbf{)}$ H $\mathbf{(}\mathbf{5}\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{)}$ J $\mathbf{(}\mathbf{-}\mathbf{6}\mathbf{,}\mathbf{2}\mathbf{)}$

Short answer

The solution to the given system of equation is option F$\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{5}\mathbf{)}$.

Step-by-step solution

Step 1. State the concept of Linear System with two variables.

A linear system of two equations with two variables is any system that can be written in the form.

$\begin{array}{l}ax+by=p\\ cx+dy=q\end{array}$

Where any of the constants can be zero with the exception that each equation must have at least one variable in it.

Also, the system is called linear if the variables are only to the first power, are only in the numerator and there are no products of variables in any of the equations.

Step 2. State the concept of ‘Substitution Method’.

Thesubstitution method is the algebraic method to solve simultaneous linear equations. As the word says, in this method, the value of one variable from one equation is substituted in the other equation. In this way, a pair of the linear equation gets transformed into one linear equation with only one variable, which can then easily be solved.

Step 3. Solve the given system of equations by substitution method.

Substitute $y=4x-7$in equation $3x-2y=-1$and simplify further to find the value of $‘x’$.

$\begin{array}{c}3x-2y=-1\\ 3x-2(4x-7)=-1\\ 3x-8x+14=-1\\ -5x+14=-1\\ -5x=-1-14\\ -5x=-15\end{array}$

Multiplying above equation throughout by $(-1)$.

$\begin{array}{c}5x=15\\ x=\frac{15}{5}\\ x=3\end{array}$

Substitute $x=3$in equation$y=4x-7$ to find the value of $‘y’$.

$\begin{array}{c}y=4\left(3\right)-7\\ y=12-7\\ y=5\end{array}$

Therefore, the solution is $x=3$and $y=5$.

Hence, option F $(3,5)$is correct.