Solving by Substitution
Solve the system using the substitution method.
Step 1 — Solve one equation for one variable. In the first equation, solve for . Add to both sides:
Divide both sides by :
Step 2 — Substitute into the other equation. Replace in the second equation with :
Step 3 — Solve the resulting one-variable equation. Distribute across the parentheses:
Combine the like terms :
Because is a true statement — and the variable has been completely eliminated — the system is consistent and the equations are dependent. The two equations represent the same line, so the system has infinitely many solutions.
This example demonstrates what happens during substitution when the two equations in a system are actually the same line in disguise: every variable term cancels out and the result is a universally true numerical statement. Recognizing as a signal of dependent equations is the algebraic counterpart of seeing two coincident lines on a graph.
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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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Internal Training: The Substitution Method Procedure
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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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An inventory manager is comparing two stock level formulas represented by the system: 4x - 3y = 6 and 15y - 20x = -30. When using the substitution method, the manager finds that the variables cancel out, leaving the true statement 0 = 0. What does this result indicate about the number of solutions for this system?
An operations manager is evaluating two different pricing models for a new service, represented by the equations 4x - 3y = 6 and 15y - 20x = -30. Upon using the substitution method to find the break-even point, the manager arrives at the statement 0 = 0. This result confirms that the equations are ____.
A retail analyst is comparing two different revenue models represented by the system: and . To determine if these models are essentially the same, the analyst uses the substitution method. Arrange the following steps of the analyst's work in the correct chronological order, from the initial variable isolation to the final interpretation of the result.
A billing coordinator is comparing two service rate models represented by the equations $4x - 3y = 6 and $15y - 20x = -30. After using the substitution method to find where the rates are equal, the coordinator arrives at the statement $0 = 0$. True or False: This result indicates that the system is inconsistent and has no possible solution.
A logistics coordinator is evaluating two shipping cost models represented by the system: and . After performing the substitution method, the coordinator identifies several properties of the system. Match each conceptual finding on the left with its corresponding mathematical description on the right.
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A data analyst is comparing two different revenue models represented by the equations $4x - 3y = 6 and $15y - 20x = -30. After using the substitution method to find where the models intersect, the analyst reaches the final statement $0 = 0$. Based on this result, what does the analyst conclude about the graphs of these two models?
An urban planner is analyzing two street alignment equations for a new development: $4x - 3y = 6 and $15y - 20x = -30. After using the substitution method and arriving at the identity $0 = 0$, the planner identifies that the two equations represent the exact same line. According to the lesson, what is the specific geometric term for lines that occupy the same space on a graph, as these two do?