Solving a System of Linear Equations by Substitution
The substitution method is an algebraic technique for solving a system of two linear equations that overcomes the limitations of graphing by producing exact answers without requiring a graph. It reduces the system to a single equation in one variable, which can then be solved using standard techniques. The procedure follows six steps:
- Solve one of the equations for either variable. Choose whichever equation and variable makes the algebra simplest — for instance, pick an equation where one variable already has a coefficient of or .
- Substitute the expression from Step 1 into the other equation. Replace the isolated variable in the second equation with the expression found in Step 1. This produces a new equation containing only one variable.
- Solve the resulting equation. Apply standard linear-equation techniques — distributing, combining like terms, and isolating the variable — to find the value of that single variable.
- Substitute the solution from Step 3 into one of the original equations to find the other variable. Plug the known value back into either original equation and solve for the second variable.
- Write the solution as an ordered pair. Express the two values in the form .
- Check that the ordered pair is a solution to both original equations. Substitute the values into each equation and verify that both produce true statements.
A useful shortcut: if one of the equations is already given in slope-intercept form (meaning one variable is already isolated on one side), then Step 1 is already complete and the process can begin directly at Step 2.
0
1
Tags
OpenStax
Elementary Algebra @ OpenStax
Ch.5 Systems of Linear Equations - Elementary Algebra @ OpenStax
Algebra
Math
Prealgebra
Intermediate Algebra @ OpenStax
Ch.4 Systems of Linear Equations - Intermediate Algebra @ OpenStax
Related
Solving a System of Linear Equations by Substitution
A small business owner is using a hand-drawn graph to compare the monthly costs of two different utility providers. Which of the following is a known practical limitation of using the graphing method to find the exact point where the costs are equal?
A professional is deciding whether to use the graphing method to solve a system of equations. Match each workplace scenario with the specific practical limitation of the graphing method it illustrates.
In a professional setting, if the intersection point of two lines on a graph falls between the grid lines (representing a non-integer solution), the graphing method is considered unreliable for finding an exact answer.
Practical Limitations of Graphing in Business Analysis
Challenges in Graph-Based Logistics Analysis
When a business analyst finds that the break-even point for two products is a decimal or fraction, the graphing method is considered unreliable because visual inspection cannot accurately identify ____ solutions.
Practical Limitations of the Graphing Method
A project manager is using a manual graph to compare the cost projections of two different construction vendors. Arrange the following steps in the order they describe the practical limitations the manager will encounter when using the graphing method.
A logistics coordinator is evaluating whether to use a manual graphing method to compare two different shipping cost models. Match each practical limitation of the graphing method with the specific challenge it presents in a professional analysis.
A logistics coordinator is using a manual graphing method to compare two shipping cost models. If the lines in the system extend well beyond a standard coordinate grid (such as a window where x and y range from -10 to 10), which practical drawback is the coordinator most likely to experience?
Learn After
Solving by Substitution
Solving by Substitution
Solving by Substitution
Solving by Substitution
Solving by Substitution
Solving by Substitution
Solving by Substitution
Solving a System of Linear Equations by Elimination
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 ).
Verifying System Solutions in Business Analysis
Internal Training: The Substitution Method Procedure
Equipment Lease Comparison
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?