\[ y=3(x+1) \] a. How many solutions does this system have? Explain your reasoning without solving the system. 4is raution Mj Alinty Solot ons bcconse has the somic integers ane the ts ily heow it nes inlinity solutions b. Based on the number of solutions, describe the graph of this system thy Siftion is beiry wilted

It appears there might be some typographical errors in your question. However, based on the provided equation and the context, I'll address the problem as follows: You provided the equation of a line: \[ y = 3(x + 1) \] which simplifies to: \[ y = 3x + 3 \] Assuming you're referring to a **system of linear equations**, let's consider a typical scenario where you're comparing this line with another linear equation to determine the number of solutions. ### a. **Number of Solutions** **Possible Scenarios:** 1. **Consistent and Independent System (One Solution):** - **Condition:** The two lines intersect at exactly one point. - **Reasoning:** They have different slopes. - **Example:** \[ y = 3x + 3 \quad \text{and} \quad y = 2x + 5 \] - **Outcome:** One unique solution. 2. **Consistent and Dependent System (Infinite Solutions):** - **Condition:** The two equations represent the same line. - **Reasoning:** They have identical slopes and y-intercepts. - **Example:** \[ y = 3x + 3 \quad \text{and} \quad y = 3x + 3 \] - **Outcome:** Infinitely many solutions. 3. **Inconsistent System (No Solution):** - **Condition:** The two lines are parallel but have different y-intercepts. - **Reasoning:** They have the same slope but different y-intercepts. - **Example:** \[ y = 3x + 3 \quad \text{and} \quad y = 3x - 2 \] - **Outcome:** No solutions. **Given your context mentioning "infinite solutions,"** it is likely that the system you're referring to falls under the **consistent and dependent** category, meaning both equations represent the **same line**. ### b. **Graph Description Based on Number of Solutions** Since the system has **infinitely many solutions**, the graphical representation would show that **both equations graph the exact same line**. This means: - **Both lines coincide completely**, lying on top of each other. - Every point on one line is also a point on the other line. - There is no distinct intersection point because the lines are identical. **Visual Representation:**  *Both lines overlap perfectly, representing infinite solutions.* ### Summary - **a.** The system has **infinitely many solutions** because both equations represent the **same line**. - **b.** Graphically, this means that **both lines coincide**, lying exactly on top of each other. If there was a different system you intended to reference, please provide the complete equations, and I'd be happy to assist further!