QQuestionMathematics
QuestionMathematics
"Solve the system of equations by any method.
3x−4y+ 2z =−15 (1)
2x+ 4y+z = 16 (2)
2x+ 3y+ 5z = 20 (3)
Enter the exact answer as an ordered triple, (x,y,z)."
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Answer
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Step 1:: Let's use the substitution method to solve this system of equations.
z = 16 - 2x - 4y \quad (4)
First, we'll solve equation (2) for z:
Step 2:: Now, we'll substitute equation (4) into equations (1) and (3) to eliminate the variable z:
3x - 4y + 2(16 - 2x - 4y) = -15
For equation (1):
Step 3:: Simplify and solve for x:
x = 12y - 47 \quad (5)
Step 4:: Now, substitute equations (4) and (5) into equation (3) to eliminate the variables x and z:
2(12y - 47) + 3y + 5(16 - 2(12y - 47) - 4y) = 20
Step 5:: Simplify and solve for y:
y = -\frac{205}{7} \approx -29.29
Step 6:: Substitute the value of y back into equation (5) to find x:
x = 12\left(-\frac{205}{7}\right) - 47 \approx -102.86
Step 7:: Substitute the values of x and y back into equation (4) to find z:
z = 16 - 2\left(-102.86\right) - 4\left(-\frac{205}{7}\right) \approx 315.14
Final Answer
The solution to the system of equations is approximately (x, y, z) = \left(- 102.86, -\frac{205}{7}, 315.14\right).
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