Solving by Elimination
To solve the system using the elimination method, follow these steps:
Step 1 — Write both equations in standard form. Both equations are already in standard form.
Step 2 — Make the coefficients of one variable opposites. To eliminate , note that the -coefficients are and . Multiply the first equation by and the second equation by so that the -coefficients become and :
The system is now
Step 3 — Add the equations to eliminate .
The -terms cancel out.
Step 4 — Solve for the remaining variable. Divide both sides by :
Step 5 — Substitute back into an original equation. Substitute into the first original equation :
Step 6 — Write the solution as an ordered pair: .
Step 7 — Check in both original equations:
- First equation: . Since is true ✓
- Second equation: . Since is true ✓
Both equations are satisfied, confirming that the solution is .
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Ch.4 Systems of Linear Equations - Intermediate Algebra @ OpenStax
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Solving by Elimination
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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
Solving by Elimination
Solving by Elimination
Solving by Elimination
Learn After
As an operations analyst resolving a supply chain discrepancy, you have isolated the unit costs of two different shipping methods into a system of linear equations: and . To update your cost models, you must solve this system. Recall the standard algebraic procedure for the elimination method by arranging the computational steps in the correct order.
In a logistics analysis, a team uses the system of equations and to balance shipping costs. When solving this system by elimination, the first equation is multiplied by 3 and the second equation is multiplied by 4 to eliminate the variable. Which of the following equations is the correct result of adding the two equations together after these multiplications?
A logistics manager is balancing resource schedules for two shipping routes, Route and Route . The resource constraints are modeled by the system: and . To solve this system using the elimination method, match each procedural step on the left with its correct mathematical result on the right.
A business analyst uses the system of linear equations and to model the break-even point for two different service plans. By applying the elimination method, the analyst determines the coordinates of the equilibrium point . Based on this model, the value of the variable at the equilibrium point is ____.
An operations analyst is balancing a resource budget using the system of linear equations and . True or False: According to the elimination method procedure for this system, the first equation is multiplied by 3 and the second equation is multiplied by 4 to create the opposite coefficients -12 and 12 for the variable .