What is the solution to the following system?
$\left\{\begin{array}{c}
4 x+3 y-z=-6 \\
6 x-y+3 z=12 \\
8 x+2 y+4 z=6
\end{array}\right.$
A. $x=1, y=-3, z=-1$
B. $x=1, y=-3, z=1$
C. $x=1, y=3, z=19$
D. $x=1, y=3, z=-2$
Answer (1)
Substitute each given option into the system of equations.
Check if all three equations are satisfied for each option.
Option 1: x = 1 , y = − 3 , z = − 1 does not satisfy the first equation.
Option 2: x = 1 , y = − 3 , z = 1 satisfies all three equations, so the solution is x = 1 , y = − 3 , z = 1 .
Explanation
Problem Analysis We are given a system of three linear equations with three unknowns, x , y , and z :
{ 4 x + 3 y − z = − 6 6 x − y + 3 z = 12 8 x + 2 y + 4 z = 6
We are also given four possible solutions and need to determine which one satisfies all three equations.
Solution Strategy We will test each option by substituting the values of x , y , and z into the equations. If all three equations are true for a given option, then that option is the solution.
Testing Option 1 Option 1: x = 1 , y = − 3 , z = − 1
Equation 1: 4 ( 1 ) + 3 ( − 3 ) − ( − 1 ) = 4 − 9 + 1 = − 4 = − 6
Since the first equation is not satisfied, we can conclude that Option 1 is not a solution.
Testing Option 2 Option 2: x = 1 , y = − 3 , z = 1
Equation 1: 4 ( 1 ) + 3 ( − 3 ) − ( 1 ) = 4 − 9 − 1 = − 6
Equation 2: 6 ( 1 ) − ( − 3 ) + 3 ( 1 ) = 6 + 3 + 3 = 12
Equation 3: 8 ( 1 ) + 2 ( − 3 ) + 4 ( 1 ) = 8 − 6 + 4 = 6
Since all three equations are satisfied, Option 2 is a solution.
Conclusion Since we have found a solution, we can stop here.
Final Answer The solution to the system of equations is x = 1 , y = − 3 , and z = 1 .
Examples
Systems of equations are used in various real-world applications, such as determining the optimal mix of products in manufacturing, balancing chemical equations, and modeling electrical circuits. For example, a company might use a system of equations to determine how many units of each product to produce in order to maximize profit, given constraints on resources such as labor and materials. Similarly, in chemistry, systems of equations are used to balance chemical reactions, ensuring that the number of atoms of each element is the same on both sides of the equation. These applications demonstrate the practical importance of understanding and solving systems of equations.
