Questions: Solve the following system of linear equations by substitution and determine whether the system has one solution, no solution, or an infinite number of solutions. If the system has one solution, find the solution. 9y = 9x - 9 -x + y = -1

Solve the following system of linear equations by substitution and determine whether the system has one solution, no solution, or an infinite number of solutions. If the system has one solution, find the solution. 9y = 9x - 9 -x + y = -1
Transcript text: Solve the following system of linear equations by substitution and determine whether the system has one solution, no solution, or an infinite number of solutions. If the system has one solution, find the solution. \[ \left\{\begin{array}{l} 9 y=9 x-9 \\ -x+y=-1 \end{array}\right. \]
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Solution

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Solution Steps

To solve the given system of linear equations using substitution, we first solve one of the equations for one variable in terms of the other. Then, substitute this expression into the other equation to find the value of one variable. Once we have one variable, substitute it back into one of the original equations to find the other variable. Finally, check if the solution satisfies both equations to determine if there is one solution, no solution, or an infinite number of solutions.

Step 1: Simplify the First Equation

The given system of equations is:

\[ \begin{cases} 9y = 9x - 9 \\ -x + y = -1 \end{cases} \]

First, simplify the first equation by dividing every term by 9:

\[ y = x - 1 \]

Step 2: Substitute into the Second Equation

Substitute \( y = x - 1 \) into the second equation:

\[ -x + (x - 1) = -1 \]

Simplify the equation:

\[ -x + x - 1 = -1 \]

This simplifies to:

\[ -1 = -1 \]

Step 3: Analyze the Result

The equation \(-1 = -1\) is always true, which means the two equations are dependent and represent the same line. Therefore, the system has an infinite number of solutions.

Final Answer

The system has an \(\boxed{\text{infinite number of solutions}}\).

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