Chapter 6.2, Problem 29HP

To determine

To identify: The difference between the solution found by graphing and by substitution method.

Explanation of Solution

For the system of equations,

  $\begin{array}{l}{a}_{1}x+b_{1}y=c_1\\ a_{2}x+b_{2}y=c_2\end{array}$

Using the substitution method, we substitute the $y$ or $x$ from one of the equations and substitute in another equation.

If we obtain an inequality after the substitution $y$ or $x$ in other equation, then the inequality indicates that the system of equations have no solution.

If we obtain an equality after the substitution $y$ or $x$ in other equation, then the equality indicates that the system of equations have infinite number of solutions.

If we obtain the value of $y$ or $x$ , from the substitution $y$ or $x$ of one equation in another equation, then it indicates the there exists unique value for $x$ and $y$ . So, the system of equation has a unique(one) solution.

Using the Graphical method, we plot both the equations(lines).

If the two lines are parallel with a finite distance between them, then it indicates the system of equations has no solution.

If the two lines are coincident, then it indicates the system of equations have infinite number of solutions.

If the two lines intersect at a point, then it indicates the system of equations has one solution.