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 :
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 the dependent case, where the same elimination produces a true statement like .
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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).
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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 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 ____.
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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'?