https://file.aiquickdraw.com/mathbot/file/1725384127458unsd230l.jpg

The image contains the following system of equations:

\[
y - 2x = 1
\]
\[
3x - 2y = 11
\]

You are asked to:
1. Graph both equations.
2. Determine if the system has a unique solution.
3. Find the solution using substitution.
4. Graph the corresponding equations.

Let's go through the steps.

### Step 1: Rewrite the equations in slope-intercept form (if needed)
For easier graphing, let's rewrite both equations in the slope-intercept form \(y = mx + b\).

1. **First equation** \(y - 2x = 1\):
   \[
   y = 2x + 1
   \]
   (This is already in slope-intercept form.)

2. **Second equation** \(3x - 2y = 11\):
   \[
   -2y = -3x + 11
   \]
   Divide by \(-2\):
   \[
   y = \frac{3}{2}x - \frac{11}{2}
   \]

### Step 2: Graph the equations
1. **First equation** \(y = 2x + 1\):
   - Slope (\(m\)) = 2
   - y-intercept (\(b\)) = 1

2. **Second equation** \(y = \frac{3}{2}x - \frac{11}{2}\):
   - Slope (\(m\)) = \(\frac{3}{2}\)
   - y-intercept (\(b\)) = \(-\frac{11}{2}\)

### Step 3: Determine the unique solution
To find the unique solution using substitution:
1. From the first equation \(y = 2x + 1\).
2. Substitute this into the second equation:
   \[
   3x - 2(2x + 1) = 11
   \]
   Simplify:
   \[
   3x - 4x - 2 = 11
   \]
   \[
   -x - 2 = 11
   \]
   \[
   -x = 13 \Rightarrow x = -13
   \]
   Substitute \(x = -13\) back into \(y = 2x + 1\):
   \[
   y = 2(-13) + 1 = -26 + 1 = -25
   \]
   So, the unique solution is \(x = -13\) and \(y = -25\).

### Step 4: Graphing the equations
If you graph both equations on the same set of axes, they will intersect at the point \((-13, -25)\).

### Conclusion
The system has a unique solution at \((-13, -25)\).

Would you like further details on any step, or have any questions?

Here are five related questions:
1. How do you determine if a system of equations has no solution, one solution, or infinitely many solutions?
2. What are the benefits of using the substitution method over the elimination method in solving systems of equations?
3. How would the solution change if the equations were parallel?
4. What is the graphical significance of the point where two lines intersect?
5. How can you check the solution of a system of equations using algebraic methods?

**Tip:** Always double-check your algebraic manipulations when solving systems of equations to avoid errors in determining the solution.

Consider the following system of equations: y - 2x = 1 and 3x - 2y = 11. Graph both equations. Does the system have a unique solution? If so, what is it? How do you solve the system by substitution and graph the corresponding equations? What do you notice?

Explore how to solve the system of linear equations y - 2x = 1 and 3x - 2y = 11. Learn the steps to graph both equations, determine if there is a unique solution, and solve the system using substitution. Ideal for students in Grades 9-11.

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