Find all solutions of the given system of equations and check your answer graphically. (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer in terms of x, where y = y(x).)
5x	 + 	6y	 = 	3
−x	 −
6
5
y	 = 	−
3
5
(x, y) =

To solve this system of equations:

\[
5x + 6y = 3
\]
\[
-x - \frac{6}{5}y = -\frac{3}{5}
\]

### Step 1: Solve for \( y \) in terms of \( x \)
Let's start by rearranging the equations to express \( y \) in terms of \( x \) if possible, or proceed by substitution or elimination.

#### First Equation
\[
5x + 6y = 3
\]

#### Second Equation
Multiply through by 5 to clear the fraction:
\[
-5x - 6y = -3
\]

Now we have the system:
1. \( 5x + 6y = 3 \)
2. \( -5x - 6y = -3 \)

### Step 2: Add the equations
Add the two equations:
\[
(5x + 6y) + (-5x - 6y) = 3 + (-3)
\]
\[
0 = 0
\]

Since adding the equations results in \(0 = 0\), this means the system is **dependent** and has infinitely many solutions.

### Step 3: Express \( y \) in terms of \( x \)
We can use one of the original equations to express \( y \) as a function of \( x \).

From the first equation:
\[
5x + 6y = 3
\]
\[
6y = 3 - 5x
\]
\[
y = \frac{3 - 5x}{6}
\]

So the solution can be written as:
\[
(x, y) = \left(x, \frac{3 - 5x}{6}\right)
\]

### Graphical Check
To confirm graphically, we would plot both lines represented by the equations. Since they are dependent, the two lines would coincide, indicating infinitely many solutions.

Would you like a detailed walkthrough of the graphical check?

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1. What does it mean for a system of equations to be dependent?
2. How can you determine if a system has infinitely many solutions without graphing?
3. What are the steps to eliminate a variable in a system of equations?
4. How can you verify the solution to a system of equations?
5. Why does a consistent system sometimes have infinitely many solutions?

**Tip:** In a dependent system, any point on one line satisfies both equations because both lines are the same.

Find all solutions of the given system of equations and check your answer graphically. (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer in terms of x, where y = y(x).) 5x + 6y = 3 −x − (6/5)y = −3/5 (x, y) =

Learn how to solve the system of linear equations 5x + 6y = 3 and −x − (6/5)y = −3/5. This problem explores algebraic techniques for solving dependent systems and verifying solutions graphically. Suitable for students in Grades 9-12.

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