Q48. - Algebra 2

Question

State the conditions for which the system below is: 

(a) consistent and dependent, 

(b) consistent and dependent, 

(c) inconsistent.

$\begin{array}{c}ax+by=c\\ dx+ey=f\end{array}$

Step-by-Step Solution

Answer

(a) The system of equations is consistent and dependent  when $\frac{a}{d}=\frac{b}{e}=\frac{c}{f}$.

(a) The system of equations is consistent and independent  when $\frac{a}{d}\ne \frac{b}{e}$.

(a) The system of equations is inconsistent when $\frac{a}{d}=\frac{b}{e}\ne \frac{c}{f}$.

(a) Step-1 – Apply the concept of consistent and inconsistent system of equations

A system of equations is said to be consistent if there exist a solution to the system.

When two lines intersect each other at one point then there is only one solution to system of equations then the system is said to be consistent and independent.

When two lines are equivalent and their graph overlap each other then there are infinite solutions to system of equations then the system is said to be consistent and dependent.

A system of equations is said to be inconsistent if there no solution to the system that is two lines are parallel to each other.

Step-2 –Interpret the conditions

Consider the system of equations.

$\begin{array}{c}ax+by=c\\ dx+ey=f\end{array}$.

The graph of the system of equations when they are consistent and dependent is provided below.

This is only possible when the coefficients of both the equations are proportional and equal to one other. Both the equations are same.

$\frac{a}{d}=\frac{b}{e}=\frac{c}{f}$

Step-3 – Interpret the system

The equations are equivalent and the same line denote the graph of each equation.

There are infinite solution to system of equations as all ordered pairs that lie on the line are the solution to system of equations.

Hence, the system of equations is consistent and dependent.

(b) Step-1 – Apply the concept of consistent and inconsistent system of equations

A system of equations is said to be consistent if there exist a solution to the system.

When two lines intersect each other at one point then there is only one solution to system of equations then the system is said to be consistent and independent.

When two lines are equivalent and their graph overlap each other then there are infinite solutions to system of equations then the system is said to be consistent and dependent.

A system of equations is said to be inconsistent if there no solution to the system that is two lines are parallel to each other.

Step-2 –Interpret the conditions

Consider the system of equations.

$\begin{array}{c}ax+by=c\\ dx+ey=f\end{array}$.

The graph of the system of equations when they are consistent and independent is provided below.

This is only possible when the coefficients of both the equations are neither proportional nor equal to one other. Both the equations are completely different.

$\frac{a}{d}\ne \frac{b}{e}$

Step-3 – Interpret the system

The equations are not equivalent and the different lines denote the graph of each equation.

The lines intersect each other at the point. There is a single point such that it satisfy the system of equations. There is an independent solution to system of equations.

Hence, the system of equations is consistent and independent.

(c) Step-1 – Apply the concept of consistent and inconsistent system of equations

A system of equations is said to be consistent if there exist a solution to the system.

When two lines intersect each other at one point then there is only one solution to system of equations then the system is said to be consistent and independent.

When two lines are equivalent and their graph overlap each other then there are infinite solutions to system of equations then the system is said to be consistent and dependent.

A system of equations is said to be inconsistent if there no solution to the system that is two lines are parallel to each other.

Step-2 –Interpret the conditions

Consider the system of equations.

$\begin{array}{c}ax+by=c\\ dx+ey=f\end{array}$.

The graph of the system of equations when they are inconsistent is provided below.

This is only possible when the coefficients of variables of both the equations are proportional and equal to one other but not proportional to number on right hand side of equal to sign. Both the equations have different numbers on right hand side.

$\frac{a}{d}=\frac{b}{e}\ne \frac{c}{f}$

Step-3 – Interpret the system

The equations are not equivalent and the different lines denote the graph of each equation.

The lines are parallel as they do not intersect each other. There is no point such that it satisfy the system of equations. There is no solution to system of equations.

Hence, the system of equations is inconsistent.