Solve the system of equations. If the system does not have one unique solution, determine the number of solutions to the system.
[tex]
\begin{aligned}
-5 x+8 y & =8 \
-2 x+5 z & =-10 \
5 y+7 z & =-9
\end{aligned}
[/tex]
Answer (2)
Solve the first equation for x : x = 5 8 y − 8 .
Substitute x into the second equation and solve for z : z = 25 16 y − 66 .
Substitute z into the third equation and solve for y : y = 1 .
Substitute y back into the equations for x and z to find x = 0 and z = − 2 . The solution is ( 0 , 1 , − 2 ) .
Explanation
Problem Analysis We are given a system of three linear equations with three unknowns: x , y , and z .
The equations are:
− 5 x + 8 y − 2 x + 5 z 5 y + 7 z = 8 = − 10 = − 9
Solution Strategy Our objective is to solve the system of equations for x , y , and z . If the system does not have a unique solution, we need to determine the number of solutions.
We can solve this system using substitution or elimination. Let's use substitution.
Solving for x First, solve the first equation for x in terms of y :
− 5 x + 8 y = 8 ⟹ − 5 x = 8 − 8 y ⟹ x = 5 8 y − 8
Substituting x into the second equation Substitute this expression for x into the second equation:
− 2 x + 5 z = − 10 ⟹ − 2 ( 5 8 y − 8 ) + 5 z = − 10
Solving for z Simplify the second equation and solve for z in terms of y :
− 5 16 y + 5 16 + 5 z = − 10 ⟹ 5 z = − 10 − 5 16 + 5 16 y ⟹ 5 z = 5 − 50 − 16 + 16 y ⟹ z = 25 16 y − 66
Substituting z into the third equation Substitute the expression for z in terms of y into the third equation:
5 y + 7 ( 25 16 y − 66 ) = − 9
Solving for y Solve for y :
5 y + 25 112 y − 462 = − 9 ⟹ 25 125 y + 112 y − 462 = − 9 ⟹ 237 y − 462 = − 225 ⟹ 237 y = 237 ⟹ y = 1
Solving for x Substitute y = 1 into the expression for x :
x = 5 8 ( 1 ) − 8 = 5 0 = 0
Solving for z Substitute y = 1 into the expression for z :
z = 25 16 ( 1 ) − 66 = 25 − 50 = − 2
Verification The solution is x = 0 , y = 1 , z = − 2 . Let's verify the solution in all three original equations:
− 5 ( 0 ) + 8 ( 1 ) − 2 ( 0 ) + 5 ( − 2 ) 5 ( 1 ) + 7 ( − 2 ) = 8 = − 10 = 5 − 14 = − 9
The solution satisfies all three equations.
Final Answer Therefore, the solution to the system of equations is x = 0 , y = 1 , and z = − 2 .
The solution set is ( 0 , 1 , − 2 ) .
Examples
Systems of equations are used in various fields, such as engineering, economics, and computer science. For example, in electrical engineering, systems of equations can be used to analyze circuits and determine the current and voltage in different parts of the circuit. In economics, they can be used to model supply and demand curves and determine equilibrium prices and quantities. In computer graphics, they are used to perform transformations and projections of 3D objects onto a 2D screen. Understanding how to solve systems of equations is crucial for solving real-world problems in these fields. For instance, consider a simple economic model where the quantity demanded Q d and the quantity supplied Q s are given by:
Q d = a − b P Q s = c + d P
where P is the price, and a , b , c , d are constants. To find the equilibrium price and quantity, we need to solve this system of equations.
The solution to the system of equations is (x, y, z) = (0, 1, -2). The equations were solved step-by-step using substitution, confirming a unique solution. All equations were verified to ensure correctness.
;
