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Solve a System of Three Linear Equations (Example 4.33)
To solve the system of equations:
3x - 4z = 0 \\ 3y + 2z = -3 \\ 2x + 3y = -5 \end{cases}$$ First, eliminate $$z$$ from the first two equations by multiplying the second equation by 2 and adding it to the first equation: $$3x - 4z = 0$$ $$6y + 4z = -6$$ Adding these equations yields a new equation with variables $$x$$ and $$y$$: $$3x + 6y = -6$$ Next, solve the new system formed by this equation and the third equation, which also contains variables $$x$$ and $$y$$. Multiply the third equation by -2 and add it to the new equation to eliminate $$y$$: $$3x + 6y = -6$$ $$-4x - 6y = 10$$ Adding these equations yields $$-x = 4$$, which simplifies to $$x = -4$$. Substitute $$x = -4$$ into the third equation to solve for $$y$$: $$2(-4) + 3y = -5$$ $$-8 + 3y = -5$$ $$3y = 3$$ $$y = 1$$ Substitute $$x = -4$$ into the first equation to solve for $$z$$: $$3(-4) - 4z = 0$$ $$-12 - 4z = 0$$ $$-4z = 12$$ $$z = -3$$ Checking the solution $$(-4, 1, -3)$$ in all three original equations confirms that it makes all three equations true.0
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Intermediate Algebra @ OpenStax
Ch.4 Systems of Linear Equations - Intermediate Algebra @ OpenStax
Algebra
Related
Identify an Inconsistent System of Three Linear Equations
Identify a Consistent System of Three Linear Equations with Dependent Equations
Solve a System of Three Linear Equations (Example 4.33)
Applications of Systems of Linear Equations with Three Variables
A logistics analyst is evaluating a potential solution, represented as an ordered triple (x, y, z), for a resource allocation system consisting of three linear equations. The analyst confirms that the values satisfy the first and second equations, but discovers they do not satisfy the third equation. Based on the definition of a solution for a system of equations, which of the following is true?
A facility manager is using a system of three linear equations to allocate a budget across three departments: Maintenance (), Operations (), and Security (). Match each mathematical term with the description that best fits its role in this budgeting scenario.
In a professional resource management system modeled by three linear equations, a specific set of values (x, y, z) is considered a solution only if it satisfies all three equations within the system.
A production supervisor at a manufacturing plant is using the elimination method to solve a system of three linear equations representing labor hours across three departments. Arrange the following procedural steps in the correct order to determine the final solution (x, y, z).
A financial analyst is evaluating a system of three linear equations to determine the optimal allocation of a corporate budget across three departments. For a specific set of values, denoted as (x, y, z), to be considered a valid solution to this model, it must satisfy all three equations. This set of three values is mathematically referred to as an ordered ____.
Validating Resource Allocation Models
Verifying Solutions in Logistics Modeling
Learn After
Solve a System of Three Linear Equations (Try It 4.65)
Solve a System of Three Linear Equations (Try It 4.66)
As a supply chain analyst, you are determining the exact number of hours three different warehouse robots (, , and ) should run to fulfill a daily quota. The operational constraints are modeled by the following system of equations:
Recalling the standard step-by-step elimination method for this type of system, what is the correct first step to begin solving it?
An inventory manager is calculating the required operational hours for three machines (, , and ) to meet a production quota. The constraints are defined by the following system:
Based on the elimination method demonstrated in the course material, arrange the following steps in the correct order to determine the values of , , and .
A transport hub manager is balancing the efficiency variances of three delivery routes: Route , Route , and Route . The operational constraints are modeled by the following system of linear equations:
Based on the step-by-step solution provided in Example 4.33, match each route variable with its correct equilibrium value.
An operations analyst is modeling the resource variances for three departments (, , and ) using the following system of linear equations:
True or False: According to the specific elimination method demonstrated in the course for this system, the first step involves eliminating the variable to produce the new linear equation .
A manufacturing manager is reviewing the operational dependencies between three assembly lines (, , and ), which are modeled by the following system of equations:
Recalling the specific steps demonstrated for this model, multiplying the second equation by 2 and adding it to the first equation eliminates the variable and results in the new two-variable equation ____.
Logistics Load Distribution Analysis
Documenting System of Equations Resolution for Operations Audit