Solving by Elimination
Solve the system using the elimination method.
Step 1 — Write both equations in standard form. The first equation is already in standard form. Rewrite the second equation by moving the -term to the left side: .
Step 2 — Clear fractions. The second equation contains the fraction . Multiply every term in the second equation by to eliminate it:
The system is now
After clearing fractions, the two equations are identical.
Step 3 — Eliminate a variable. Multiply the second equation by and add:
Adding both sides:
Because is a true statement and both variables have been completely eliminated, the equations are consistent and dependent. Their graphs are coincident lines — both equations describe the same line. The system has infinitely many solutions.
This example illustrates a key diagnostic for the elimination method: when the process of eliminating variables produces a universally true numerical statement like , the two equations are dependent. In this case, recognizing that the two equations became identical after clearing fractions is an additional clue that the lines coincide.
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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).
Final Verification in Labor Cost Analysis
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?
Choosing the Most Convenient Method for and
Solving by Elimination
Solving by Elimination
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Solving by Elimination
Learn After
A production manager is using the elimination method to solve a system of equations representing two different assembly line costs: 3x + 4y = 12 and y = 3 - (3/4)x. After clearing fractions and adding the equations, the manager arrives at the statement 0 = 0. What does this result indicate about the relationship between the two cost models?
A financial analyst is comparing two debt-to-income models represented by the equations 3x + 4y = 12 and y = 3 - 0.75x. Match the following terms to their correct description based on the result of solving this system by elimination.
A logistics coordinator is comparing two inventory formulas: and . After using the elimination method to solve the system, the coordinator reaches the numerical statement . True or False: This result indicates that the system is inconsistent and has no solution.
A business analyst is evaluating two different cost models represented by the equations $3x + 4y = 12y = 3 - \frac{3}{4}x$. To solve this system using the elimination method and determine their relationship, arrange the following procedural steps in the correct order.
Interpreting System Results in Utility Management
Classification of Redundant Operations Models
A supply chain analyst is comparing two logistics cost models: and . After converting the second model to standard form and clearing the fractions, the analyst discovers that both equations are identical. When applying the elimination method to solve the system, the variables are completely eliminated, resulting in the true statement . This result indicates that the system has ____ solutions.
Logistics Model Classification
A logistics analyst is comparing two different cost functions for transportation: $3x + 4y = 12y = 3 - \frac{3}{4}xAx + By = C$) with integer coefficients. Which of the following represents the correct standard form for the second function?
A logistics coordinator is reviewing two different delivery route cost models: $3x + 4y = 12y = 3 - \frac{3}{4}x$. After simplifying the equations and using the elimination method, the coordinator discovers that the two equations represent the exact same line. What is the mathematical term for lines that lie directly on top of each other and share all points?