Solving by Substitution
Solve the system using the substitution method. The second equation is already solved for , so Step 1 is already complete. Step 2 — Substitute into the other equation. Replace in the first equation with : Step 3 — Solve the resulting one-variable equation. Simplify the substituted term to get : , which simplifies to . Because is a false statement — and the variable has been completely eliminated — the equations are inconsistent. The two lines are parallel and never intersect, so the system has no solution. This example illustrates the algebraic indicator of an inconsistent system: when the substitution process eliminates all variables and produces a false numerical statement such as , no ordered pair can satisfy both equations simultaneously. This contrasts with a dependent system, where the substitution process produces a true numerical statement like .
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A small business owner is comparing the monthly costs of two different utility providers using a system of linear equations. To find the exact usage point where the costs are equal using the substitution method, arrange the following procedural steps in the correct order from start to finish.
A project manager is comparing the costs of two different software subscriptions using a system of linear equations. To solve this system using the substitution method, what is the first step the manager should take if neither equation has a variable already isolated?
A logistics coordinator is comparing two different fuel supply contracts whose costs are modeled by a system of linear equations. Match each step of the substitution method with its corresponding action in this business analysis.
A project manager is comparing two different project estimates using a system of linear equations. To solve this system using the substitution method, the manager substitutes an expression for one variable into the second equation. This step is designed to produce a new equation that contains exactly ____ variable(s).
When using the substitution method to solve a system of linear equations for an office lease comparison, the first step of solving for one variable is mandatory even if one equation is already provided in a form where a variable is isolated (such as ).
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A logistics coordinator is comparing two different fuel supply contracts whose costs are modeled by a system of linear equations. According to the standard definition of the substitution method, what is the primary benefit of using this algebraic technique instead of the graphing method?
A facilities manager is using the substitution method to solve a system of linear equations representing the monthly costs of two different security contracts. According to the standard efficiency guidelines for this method, which characteristic should the manager look for when choosing which variable to isolate in the first step?
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A production manager uses the substitution method to solve a system of equations representing two different assembly line speeds. The algebraic process leads to the statement 0 = -10. Based on this result, what can the manager conclude about the assembly lines?
A logistics manager uses the substitution method to solve the system {5x - 2y = -10, y = (5/2)x} to find where two shipping costs are equal. After substituting, the manager arrives at the statement 0 = -10. Because this is a false statement, the manager concludes that the system is ____.
A corporate logistics manager is evaluating two delivery routes using the system of equations . Match each algebraic or geometric component with its correct description according to the substitution method.
A logistics planner is evaluating two delivery routes with cost functions represented by the system {5x - 2y = -10, y = (5/2)x}. Arrange the following algebraic steps in the correct order to solve this system using the substitution method and determine if the routes will ever share a common cost point.
A business analyst is comparing two linear revenue models using the system {5x - 2y = -10, y = (5/2)x}. True or False: If the substitution method leads to the false numerical statement 0 = -10, this result indicates that the system is dependent and has infinitely many solutions.
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A business analyst is comparing two cost models represented by the equations $5x - 2y = -10y = \frac{5}{2}x. When attempting to find an intersection point using the substitution method, the analyst arrives at the false statement $0 = -10. Which geometric description correctly identifies the relationship between these two models?
A business analyst is using the substitution method to determine if two revenue models, represented by the system , will ever reach an intersection point. After the variables are eliminated during the simplification step, which specific algebraic outcome identifies this system as 'inconsistent'?