Solve the system by the method of substitution. See Example...

Key Concepts

Verification of Solutions

Verification of solutions entails substituting the values found for the variables back into each of the original equations. This step confirms that the solutions satisfy all equations in the system, thereby ensuring that no errors occurred during the manipulation and solution process.

Isolating a Variable

Isolating a variable involves rearranging an equation so that one variable appears alone on one side of the equation. This step is crucial in the substitution method because it allows for the clear expression of one variable in terms of another before it is substituted into the subsequent equations.

Back-Substitution

Back-substitution is the process of substituting the value of a variable, which has been determined from one of the equations, back into another equation to solve for the remaining variables. This sequential approach ensures that all variables in the system are resolved accurately.

Systems of Linear Equations

A system of linear equations is a set of equations where each equation is linear, meaning each term is either a constant or the product of a constant and a single variable. The goal is to find a common set of values for the variables that simultaneously satisfies all the equations. This concept is fundamental in algebra and is applied in various fields, including physics, engineering, and economics.

Substitution Method

The substitution method is a technique used to solve systems of equations by expressing one variable in terms of another using one of the equations. This expression is then substituted into the other equation(s), effectively reducing the system to a single equation with one variable. This method simplifies the process of finding the solution by breaking it into more manageable steps.

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