https://r2.erweima.ai/mathbot/image/1725930524133kf5yl2ol.jpg

To solve the systems of linear equations in the image step by step, we need to graph them and find where the lines intersect. This will give us the solution to the system. Let's go over each system of equations:

### System (a):
\[
\begin{aligned}
1. &\ x + y = 2 \\
2. &\ 3x + 3y = 5
\end{aligned}
\]

### Step 1: Express each equation in slope-intercept form (y = mx + b).

1. **Equation 1: \( x + y = 2 \)**
   \[
   y = -x + 2
   \]
   This is in slope-intercept form with slope \( -1 \) and y-intercept \( 2 \).

2. **Equation 2: \( 3x + 3y = 5 \)**
   \[
   3y = -3x + 5 \quad \text{(subtract 3x from both sides)} \\
   y = -x + \frac{5}{3} \quad \text{(divide everything by 3)}
   \]
   This is also in slope-intercept form with slope \( -1 \) and y-intercept \( \frac{5}{3} \approx 1.67 \).

### Step 2: Graph the two equations.

1. For \( y = -x + 2 \):
   - The y-intercept is \( (0, 2) \).
   - The slope is \( -1 \), so for every step to the right (positive x direction), the graph goes down by 1 unit.
   
2. For \( y = -x + \frac{5}{3} \):
   - The y-intercept is \( (0, \frac{5}{3}) \approx (0, 1.67) \).
   - The slope is also \( -1 \), so it follows the same pattern (goes down by 1 unit for every step to the right).

### Step 3: Analyze the graphs.
The two lines have the same slope but different y-intercepts. This means they are parallel and will never intersect. Therefore, **there is no solution** to this system of equations. The system is inconsistent.

---

### System (b):
\[
\begin{aligned}
1. &\ x + y = 1 \\
2. &\ 3x + 2y = 3
\end{aligned}
\]

### Step 1: Express each equation in slope-intercept form.

1. **Equation 1: \( x + y = 1 \)**
   \[
   y = -x + 1
   \]
   This is in slope-intercept form with slope \( -1 \) and y-intercept \( 1 \).

2. **Equation 2: \( 3x + 2y = 3 \)**
   \[
   2y = -3x + 3 \quad \text{(subtract 3x from both sides)} \\
   y = -\frac{3}{2}x + \frac{3}{2} \quad \text{(divide everything by 2)}
   \]
   This is also in slope-intercept form with slope \( -\frac{3}{2} \) and y-intercept \( \frac{3}{2} \approx 1.5 \).

### Step 2: Graph the two equations.

1. For \( y = -x + 1 \):
   - The y-intercept is \( (0, 1) \).
   - The slope is \( -1 \), so for every step to the right, the graph goes down by 1 unit.

2. For \( y = -\frac{3}{2}x + \frac{3}{2} \):
   - The y-intercept is \( (0, \frac{3}{2}) = (0, 1.5) \).
   - The slope is \( -\frac{3}{2} \), meaning for every 2 units to the right, the graph goes down by 3 units.

### Step 3: Find the intersection point.
By graphing these two lines, we can find that they intersect at one point. Let's solve this algebraically by setting the equations equal to each other:
\[
-x + 1 = -\frac{3}{2}x + \frac{3}{2}
\]
Multiply the entire equation by 2 to eliminate fractions:
\[
-2x + 2 = -3x + 3
\]
Solve for \( x \):
\[
x = 1
\]
Now substitute \( x = 1 \) into one of the original equations to solve for \( y \):
\[
1 + y = 1 \quad \Rightarrow \quad y = 0
\]
So, the solution to this system is \( (1, 0) \).

---

I can continue with the remaining systems (c) and (d) if you would like! Would you like more details or do you have any specific questions?

Here are 5 questions to extend this discussion:
1. How do you determine if a system has no solution graphically?
2. Why are parallel lines significant when solving systems of equations?
3. What is the significance of the slope in the context of graphing linear equations?
4. How would you solve a system algebraically if it’s not easily graphable?
5. What are the different types of solutions a system of equations can have?

**Tip:** Parallel lines always indicate that the system of equations has no solution, as they never intersect.

Determina gráficamente la solución de los siguientes sistemas:
 a) \(x + y = 2\) , \(3x + 3y = 5\)
 b) \(x + y = 1\) , \(3x + 2y = 3\)
 c) \(2x + y = 5\) , \(x + 3y = 5\)
 d) \(x - y = 1\) , \(2x + y = 8\)

Learn how to solve systems of linear equations graphically. This problem covers key concepts such as slope-intercept form and how to find the intersection points of lines. Includes step-by-step solutions for systems like x + y = 2 and 3x + 3y = 5. Suitable for students in Grades 8-10.

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