What is the solution to the system of equations?
$\left\{\begin{array}{r}
-3 x-3 y+2 z=-7 \\
z=1 \\
-2 x-3 y+z=-6
\end{array}\right.$
A. $(2,1,1)$
B. $(2,1,-1)$
C. $(-2,1,1)$
D. $(2,-1,1)$
Answer (1)
Substitute z = 1 into the first and third equations.
Simplify the equations to − 3 x − 3 y = − 9 and − 2 x − 3 y = − 7 .
Solve for x in terms of y : x = 3 − y .
Substitute x into the second equation and solve for y : y = 1 .
Substitute y back to find x : x = 2 . The solution is ( 2 , 1 , 1 ) .
Explanation
Understanding the Problem We are given a system of three equations with three unknowns: x , y , and z .
The equations are:
− 3 x − 3 y + 2 z = − 7 z = 1 − 2 x − 3 y + z = − 6
We want to find the values of x , y , and z that satisfy all three equations simultaneously.
Substituting z = 1 Substitute z = 1 into the first and third equations to eliminate z . The first equation becomes − 3 x − 3 y + 2 ( 1 ) = − 7 which simplifies to − 3 x − 3 y = − 9 The third equation becomes − 2 x − 3 y + 1 = − 6 which simplifies to − 2 x − 3 y = − 7
Simplifying the Equations Now we have a system of two equations with two unknowns: − 3 x − 3 y = − 9 − 2 x − 3 y = − 7 Divide the first equation by − 3 to simplify it to x + y = 3 So, x = 3 − y
Solving for y Substitute x = 3 − y into the second equation: − 2 ( 3 − y ) − 3 y = − 7 Simplify and solve for y :
− 6 + 2 y − 3 y = − 7 − y = − 1 y = 1
Solving for x Substitute y = 1 back into x = 3 − y to find x :
x = 3 − 1 = 2
Final Answer The solution is x = 2 , y = 1 , and z = 1 . Therefore, the solution is ( 2 , 1 , 1 ) .
Examples
Systems of equations are used in various fields, such as economics, engineering, and computer science. For example, in economics, systems of equations can be used to model the supply and demand of goods in a market. Each equation represents a relationship between the price and quantity of a particular good, and the solution to the system represents the equilibrium price and quantity where supply equals demand. Understanding how to solve these systems allows economists to predict market outcomes and analyze the effects of different policies.
