https://file.aiquickdraw.com/mathbot/file/1725503642269o1tg7uya.jpg

The image contains a set of graphing problems related to linear equations and inequalities. Let's go through each of the problems one by one:

---

**1. Graph the equation using a table:**

   \[ 4x - 1 = y \]

   - To graph this, you can create a table of values for \( x \) and compute the corresponding values for \( y \). 
   
   Example:

   \[
   \begin{array}{c|c}
   x & y \\
   \hline
   0 & -1 \quad (\text{from } 4(0) - 1 = -1) \\
   1 & 3 \quad (\text{from } 4(1) - 1 = 3) \\
   2 & 7 \quad (\text{from } 4(2) - 1 = 7) \\
   \end{array}
   \]

   Plot these points and connect them to form a straight line.

---

**2. Graph the equation using a table:**

   \[ 3 = 5x - y \]
   
   Rearranging it into slope-intercept form \( y = 5x - 3 \):

   - You can similarly create a table of values for \( x \) and calculate \( y \).
   
   Example:

   \[
   \begin{array}{c|c}
   x & y \\
   \hline
   0 & -3 \quad (\text{from } y = 5(0) - 3) \\
   1 & 2 \quad (\text{from } y = 5(1) - 3) \\
   2 & 7 \quad (\text{from } y = 5(2) - 3) \\
   \end{array}
   \]

   Plot these points and connect them to form a line.

---

**3. Graph each equation using the X and Y intercepts:**

   **(a) \( 3y - x = 6 \)**

   - To find the x-intercept, set \( y = 0 \) and solve for \( x \):
     
     \[
     0 - x = 6 \quad \Rightarrow \quad x = -6
     \]
   
   - To find the y-intercept, set \( x = 0 \) and solve for \( y \):
     
     \[
     3y - 0 = 6 \quad \Rightarrow \quad y = 2
     \]

   Plot the points \( (-6, 0) \) and \( (0, 2) \), then draw the line through them.

   **(b) \( 2x - 2y = 6 \)**

   - For the x-intercept, set \( y = 0 \):

     \[
     2x = 6 \quad \Rightarrow \quad x = 3
     \]

   - For the y-intercept, set \( x = 0 \):

     \[
     -2y = 6 \quad \Rightarrow \quad y = -3
     \]

   Plot the points \( (3, 0) \) and \( (0, -3) \), and draw the line through them.

---

**4. Graph the inequality:**

   \[ y + 1 \geq 2x \]

   - First, rewrite the inequality as \( y \geq 2x - 1 \).
   - To graph the boundary line \( y = 2x - 1 \), plot points as you did for equations.
   - Then shade the region above the line, as the inequality is \( y \geq 2x - 1 \), which includes the points where \( y \) is greater than or equal to the line.

---

Let me know if you need further explanation or details on any specific part! Here are five related questions for practice:

1. What is the slope of the line \( y = 4x - 1 \)?
2. How do you find the slope-intercept form of \( 5x - y = 3 \)?
3. What are the intercepts of the equation \( 3x + 2y = 6 \)?
4. How would you graph the inequality \( y \leq -x + 3 \)?
5. Can you find the slope and y-intercept of the equation \( 2x - y = 5 \)?

**Tip**: When graphing inequalities, always remember to use a dashed line for strict inequalities (like \( y > 2x \)) and a solid line for non-strict ones (like \( y \geq 2x \)).

Graph each equation and inequality using tables, x and y intercepts, and shading regions.

Learn how to graph linear equations using tables, find x and y intercepts, and graph inequalities. This guide covers step-by-step solutions for problems like 4x - 1 = y and y + 1 ≥ 2x, making it suitable for students in Grades 7-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation