Use the substitution or elimination method to solve each system of equations. Identify any inconsistent systems or systems with infinitely many solutions. If a system has infinitely many solutions, write the solution set with y arbitrary.
2x + 6y = 6
5x + 9y = 9

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Start by writing down the system of equations: \[2x + 6y = 6\] \[5x + 9y = 9\]

Choose a method to solve the system. Here, we will use the elimination method. To do this, multiply each equation by a suitable number so that the coefficients of one variable are opposites. For example, multiply the first equation by 5 and the second equation by 2 to align the coefficients of \(x\): \[5(2x + 6y) = 5(6) \Rightarrow 10x + 30y = 30\] \[2(5x + 9y) = 2(9) \Rightarrow 10x + 18y = 18\]

Subtract the second new equation from the first to eliminate \(x\): \[(10x + 30y) - (10x + 18y) = 30 - 18\] This simplifies to: \[12y = 12\]

Solve for \(y\) by dividing both sides by 12: \[y = \frac{12}{12}\]

Substitute the value of \(y\) back into one of the original equations to solve for \(x\). For example, substitute into the first equation: \[2x + 6\left(y\right) = 6\] Then solve for \(x\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Solutions can be unique, infinite, or nonexistent depending on the relationships between the equations.

Substitution and Elimination Methods

Substitution involves solving one equation for a variable and substituting it into the other equation to find the solution. Elimination involves adding or subtracting equations to eliminate one variable, simplifying the system to a single equation. Both methods are systematic approaches to solve systems of linear equations.

Types of Solutions: Consistent, Inconsistent, and Dependent Systems

A consistent system has at least one solution; it can be unique or infinite. An inconsistent system has no solution, often due to parallel lines. Dependent systems have infinitely many solutions, where equations represent the same line. Identifying these types helps in interpreting the solution set correctly.

Use the substitution or elimination method to solve each system of equations. Identify any inconsistent systems or systems with infinitely many solutions. If a system has infinitely many solutions, write the solution set with y arbitrary. 2x + 6y = 6 5x + 9y = 9