Solving by Elimination
Solve the system using the elimination method.
Step 1 — Write both equations in standard form. Both equations are already in the form .
Step 2 — Make the coefficients of one variable opposites. To eliminate , multiply the second equation by so that the -coefficients become and :
The system becomes
Step 3 — Add the equations.
Because is a false statement and both variables have been completely eliminated, the equations are inconsistent. Their graphs are parallel lines that never intersect. The system has no solution.
This outcome is the opposite of the dependent case: when all variable terms cancel during elimination and the resulting numerical statement is false (such as ), no ordered pair can satisfy both equations simultaneously. When the statement is true (such as ), the system has infinitely many solutions instead.
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Strategy for Choosing the Most Convenient Method to Solve a System of Linear Equations
A retail manager is using a system of linear equations to determine the unit cost of two different products based on bulk invoices. To solve this system using the elimination method, the manager must follow a specific sequence. Arrange the steps below in the correct order.
A production manager is using the elimination method to solve a system of linear equations representing the costs of two different raw materials. According to the standard steps of this method, what is the primary goal of multiplying one or both equations by a specific constant?
A logistics analyst at a global shipping company is using systems of linear equations to optimize delivery routes and fuel costs. When applying the elimination method, the analyst must correctly identify the technical requirements of each step. Match each term below with its corresponding role or definition in the elimination process.
A budget analyst is using the elimination method to determine the unit costs of two different service contracts. To eliminate a variable by adding the two equations together, the analyst must first ensure that the coefficients of the chosen variable are _________ (for example, +8 and -8).
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When a budget analyst uses the elimination method to solve a system of linear equations—such as those used to compare the costs of two different service contracts—the primary requirement for removing a variable is to ensure its coefficients in both equations are opposites (for example, +10 and -10) before the equations are added together.
Procedure for Elimination in Cost Analysis
Standard Operating Procedure for the Elimination Method in Cost Analysis
An operations analyst is preparing to use the elimination method to solve a system of linear equations representing budget allocations. According to the standard 7-step procedure for this method, what is the required first step the analyst must take with the equations before attempting to eliminate a variable?
A data analyst at a logistics company is using the elimination method to solve a system of linear equations representing fuel costs (f) and labor costs (l). After the analyst adds the equations together to successfully eliminate the fuel cost variable (f), which of the following best describes the resulting algebraic form they must solve next?
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Learn After
A project manager is evaluating two different production constraints represented by the system of equations: -6x + 15y = 10 and 2x - 5y = -5. After multiplying the second equation by 3 and adding it to the first to eliminate the variables, the manager arrives at the statement 0 = -5. What does this result indicate about the production constraints?
A logistics coordinator is modeling two delivery routes using the system of equations -6x + 15y = 10 and 2x - 5y = -5. After using the elimination method, the variables cancel out to produce the statement 0 = -5. Because this statement is false, the system is classified as ____.
A logistics coordinator is analyzing two delivery routes modeled by the following system of linear equations:
After using the elimination method (by multiplying the second equation by 3 and adding it to the first), the coordinator reaches several mathematical conclusions. Match each term below with its correct description for this specific system.
An operations analyst is evaluating two conflicting production constraints modeled by the system of equations {-6x + 15y = 10, 2x - 5y = -5}. After using the elimination method to reach the mathematical statement 0 = -5, the analyst concludes that the system is inconsistent and has no solution. Is this conclusion correct?
A supply chain analyst is comparing two cost functions modeled by the system: and . The analyst intends to find a common solution using the elimination method. Arrange the following steps in the correct order to reflect the mathematical process and the final conclusion reached.
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A budget analyst is evaluating two financial constraints modeled by the following system of equations:
When using the elimination method, the analyst multiplies the second equation by 3 and adds it to the first equation. Which specific mathematical statement is produced by this addition to show that the system has no solution?
A technical support specialist is testing a math application using the system and . When the specialist multiplies the second equation by 3 and adds it to the first to eliminate the variable, what occurs with the variables in the resulting statement?