What is the solution to the system of equations?
$\left\{\begin{aligned}
-x+2 y+z & =10 \\
z & =6 \\
3 x-2 y+2 z & =8
\end{aligned}\right.$
A. $(0,2,6)$
B. $(0,-2,6)$
C. $(0,2,-6)$
D. $(-1,2,6)$
Answer (2)
Substitute z = 6 into the first and third equations.
Simplify the equations to − x + 2 y = 4 and 3 x − 2 y = − 4 .
Solve for x by adding the two equations: 2 x = 0 , so x = 0 .
Substitute x = 0 into − x + 2 y = 4 to find y = 2 . The solution is ( 0 , 2 , 6 ) .
Explanation
Analyze the problem We are given a system of three equations with three variables: x, y, and z. Our goal is to find the values of x, y, and z that satisfy all three equations simultaneously. The equations are:
-x + 2y + z = 10
z = 6
3x - 2y + 2z = 8
Substitute z = 6 We can substitute the value of z from the second equation (z = 6) into the first and third equations to reduce the system to two equations with two variables (x and y).
Substituting z = 6 into the first equation: -x + 2y + 6 = 10 -x + 2y = 4
Substituting z = 6 into the third equation: 3x - 2y + 2(6) = 8 3x - 2y + 12 = 8 3x - 2y = -4
Solve for x Now we have a system of two equations with two variables:
-x + 2y = 4
3x - 2y = -4
We can solve this system by adding the two equations to eliminate the variable y: (-x + 2y) + (3x - 2y) = 4 + (-4) 2x = 0 x = 0
Solve for y Now that we have the value of x (x = 0), we can substitute it back into either of the two equations to solve for y. Let's use the first equation:
-x + 2y = 4 -0 + 2y = 4 2y = 4 y = 2
Verify the solution We have found x = 0, y = 2, and we know z = 6. Therefore, the solution to the system of equations is (0, 2, 6).
To verify the solution, substitute these values into the original equations:
-x + 2y + z = -0 + 2(2) + 6 = 0 + 4 + 6 = 10 (Correct)
z = 6 (Correct)
3x - 2y + 2z = 3(0) - 2(2) + 2(6) = 0 - 4 + 12 = 8 (Correct)
The solution (0, 2, 6) satisfies all three equations.
State the final answer The solution to the system of equations is (0, 2, 6).
Examples
Systems of equations are used in various real-world applications, such as determining the optimal mix of products to maximize profit, balancing chemical equations, and analyzing electrical circuits. For instance, a company might use a system of equations to determine how many units of each product they need to sell to reach a specific revenue target, considering production costs and market demand. By solving the system, they can find the exact quantities that satisfy all the constraints and achieve their goal.
The solution to the system of equations is (0, 2, 6), which can be verified by substituting the values back into the original equations. This confirms that the chosen answer is correct. The correct option is A: (0, 2, 6).
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