Find the solution set of the following linear system. Identify inconsistent systems and dependent equations.
$\left\{\begin{aligned} 4 a+7 b & =-11 \\ 8 a+2 c & =2 \\ 6 b+2 c & =4 \end{aligned}\right.$
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A. The system is consistent and the solution set is \$\{(\square \cdot \square)\}$.
B. The system is consistent and dependent. The solution set is the set of all ordered triples \$\{(\square, \square, c)\}$, where c is any real number.
C. The system is inconsistent.
Answer (1)
Solve equation 2 for c: c = 1 − 4 a .
Substitute c into equation 3: 3 b − 4 a = 1 .
Solve the system of two equations with two variables a and b, 4 a + 7 b = − 11 and − 4 a + 3 b = 1 , by adding the equations to eliminate a, which gives b = − 1 .
Substitute b = -1 to find a = -1, and then find c = 5. The solution is ( − 1 , − 1 , 5 ) .
Explanation
Understanding the Problem We are given a system of three linear equations with three unknowns: a, b, and c.
Listing the Equations The equations are:
4 a + 7 b = − 11
8 a + 2 c = 2
6 b + 2 c = 4
Stating the Objective Objective: Find the solution set of the given linear system and determine if the system is inconsistent or dependent.
Planning the Solution Solve the system of equations using either substitution or elimination.
Solving for c in Equation (2) First, solve equation (2) for c: 2 c = 2 − 8 a ⟹ c = 1 − 4 a .
Substituting c into Equation (3) Substitute this expression for c into equation (3): 6 b + 2 ( 1 − 4 a ) = 4 ⟹ 6 b + 2 − 8 a = 4 ⟹ 6 b − 8 a = 2 ⟹ 3 b − 4 a = 1 .
Two Equations with Two Variables Now we have two equations with two variables a and b:
4 a + 7 b = − 11
− 4 a + 3 b = 1
Eliminating a and Solving for b Add the two equations to eliminate a: 10 b = − 10 ⟹ b = − 1 .
Solving for a Substitute b = -1 into equation (1): 4 a + 7 ( − 1 ) = − 11 ⟹ 4 a − 7 = − 11 ⟹ 4 a = − 4 ⟹ a = − 1 .
Solving for c Substitute a = -1 into the expression for c: c = 1 − 4 ( − 1 ) = 1 + 4 = 5 .
Stating the Solution The solution is a = -1, b = -1, c = 5. The system is consistent and has a unique solution. The solution set is {(-1, -1, 5)}.
Examples
Linear systems appear in many engineering and science applications. For example, when designing electrical circuits, the currents in different branches can be found by solving a system of linear equations. Similarly, in chemical engineering, balancing chemical reactions involves solving a system of linear equations to ensure mass conservation. Understanding how to solve these systems is crucial for analyzing and designing such systems effectively.
