Use a calculator to solve the system for variable x.
[tex]\begin{array}{c}
5 x+3 y-z=34 \\
x-3 y+5 z=50 \\
2 x+y+z=26
\end{array}[/tex]
A. x=3
B. x=7
C. x=8
D. x=1
Answer (1)
Substitute each possible value of x into the system of equations.
For x = 7, solve for y and z and check if the values satisfy all three equations. The solution is inconsistent.
For x = 8, solve for y and z and check if the values satisfy all three equations. The solution is consistent with y=1 and z=9.
Conclude that the correct value for x is 8 .
Explanation
Analyze the problem We are given a system of three linear equations with three variables x, y, and z. Our goal is to find the value of x. The given equations are:
Equation 1: 5 x + 3 y − z = 34 Equation 2: x − 3 y + 5 z = 50 Equation 3: 2 x + y + z = 26
We are also given four possible values for x: 3, 7, 8, and 1. We need to determine which of these values is the correct solution for x in the system of equations.
Test x=7 Let's test each of the given values for x to see which one satisfies the system of equations. We will substitute each value into the equations and solve for y and z. If we find consistent values for y and z that satisfy all three equations, then that value of x is the correct solution.
Case 1: x = 7 Substitute x = 7 into the equations: Equation 1: 5 ( 7 ) + 3 y − z = 34 ⇒ 35 + 3 y − z = 34 ⇒ 3 y − z = − 1 Equation 2: 7 − 3 y + 5 z = 50 ⇒ − 3 y + 5 z = 43 Equation 3: 2 ( 7 ) + y + z = 26 ⇒ 14 + y + z = 26 ⇒ y + z = 12
Now we have a system of three equations with two variables: 3 y − z = − 1 − 3 y + 5 z = 43 y + z = 12
Adding the first two equations, we get: 4 z = 42 ⇒ z = 4 42 = 2 21 = 10.5 Substituting z = 10.5 into the third equation: y + 10.5 = 12 ⇒ y = 1.5 Now, let's check if these values satisfy the first equation: 3 ( 1.5 ) − 10.5 = 4.5 − 10.5 = − 6 = − 1 . So, x = 7 is not a solution.
Test x=8 Case 2: x = 8 Substitute x = 8 into the equations: Equation 1: 5 ( 8 ) + 3 y − z = 34 ⇒ 40 + 3 y − z = 34 ⇒ 3 y − z = − 6 Equation 2: 8 − 3 y + 5 z = 50 ⇒ − 3 y + 5 z = 42 Equation 3: 2 ( 8 ) + y + z = 26 ⇒ 16 + y + z = 26 ⇒ y + z = 10
Now we have a system of three equations with two variables: 3 y − z = − 6 − 3 y + 5 z = 42 y + z = 10
Adding the first two equations, we get: 4 z = 36 ⇒ z = 9 Substituting z = 9 into the third equation: y + 9 = 10 ⇒ y = 1 Now, let's check if these values satisfy the first equation: 3 ( 1 ) − 9 = 3 − 9 = − 6 . So, x = 8 is a solution. Let's verify that x=8, y=1, and z=9 satisfy all three original equations: Equation 1: 5 ( 8 ) + 3 ( 1 ) − 9 = 40 + 3 − 9 = 34 (Correct) Equation 2: 8 − 3 ( 1 ) + 5 ( 9 ) = 8 − 3 + 45 = 50 (Correct) Equation 3: 2 ( 8 ) + 1 + 9 = 16 + 1 + 9 = 26 (Correct) Therefore, x = 8 is the correct solution.
Conclusion Since we found that x = 8 satisfies all three equations, we can conclude that x = 8 is the solution to the system of equations.
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. In economics, systems of equations can be used to model supply and demand curves to find equilibrium prices and quantities. Understanding how to solve systems of equations is crucial for making informed decisions in many fields.
