Give me some problems with paraelle and perpendicular lines with step by step answers right under (giving me a y coordinate to go through)

Here are some problems involving parallel and perpendicular lines. Each problem includes a specific \( y \)-coordinate and step-by-step solutions.

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### **Problem 1: Find the equation of a line parallel to \( y = 2x + 3 \) that passes through the point \( (4, 6) \).**

#### **Solution:**
1. A parallel line has the same slope. The slope of \( y = 2x + 3 \) is \( m = 2 \).
2. The equation of a line is \( y = mx + b \). Substitute \( m = 2 \) and \( (x, y) = (4, 6) \):
   \[
   6 = 2(4) + b
   \]
3. Solve for \( b \):
   \[
   6 = 8 + b \implies b = -2
   \]
4. The equation of the line is:
   \[
   y = 2x - 2
   \]

---

### **Problem 2: Find the equation of a line perpendicular to \( y = -\frac{1}{3}x + 5 \) that passes through \( (6, -2) \).**

#### **Solution:**
1. A perpendicular line has a slope that is the negative reciprocal. The slope of \( y = -\frac{1}{3}x + 5 \) is \( m = -\frac{1}{3} \), so the perpendicular slope is:
   \[
   m = 3
   \]
2. Using \( y = mx + b \), substitute \( m = 3 \) and \( (x, y) = (6, -2) \):
   \[
   -2 = 3(6) + b
   \]
3. Solve for \( b \):
   \[
   -2 = 18 + b \implies b = -20
   \]
4. The equation of the perpendicular line is:
   \[
   y = 3x - 20
   \]

---

### **Problem 3: Find the equation of a line parallel to \( y = \frac{5}{4}x - 7 \) that passes through \( (2, 3) \).**

#### **Solution:**
1. A parallel line has the same slope. The slope is \( m = \frac{5}{4} \).
2. Using \( y = mx + b \), substitute \( m = \frac{5}{4} \) and \( (x, y) = (2, 3) \):
   \[
   3 = \frac{5}{4}(2) + b
   \]
3. Simplify and solve for \( b \):
   \[
   3 = \frac{10}{4} + b \implies 3 = 2.5 + b \implies b = 0.5
   \]
4. The equation of the line is:
   \[
   y = \frac{5}{4}x + 0.5
   \]

---

### **Problem 4: Find the equation of a line perpendicular to \( y = -2x + 1 \) that passes through \( (-3, -4) \).**

#### **Solution:**
1. A perpendicular line has a slope that is the negative reciprocal. The slope of \( y = -2x + 1 \) is \( m = -2 \), so the perpendicular slope is:
   \[
   m = \frac{1}{2}
   \]
2. Using \( y = mx + b \), substitute \( m = \frac{1}{2} \) and \( (x, y) = (-3, -4) \):
   \[
   -4 = \frac{1}{2}(-3) + b
   \]
3. Simplify and solve for \( b \):
   \[
   -4 = -\frac{3}{2} + b \implies b = -4 + \frac{3}{2} \implies b = -\frac{5}{2}
   \]
4. The equation of the perpendicular line is:
   \[
   y = \frac{1}{2}x - \frac{5}{2}
   \]

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### **Problem 5: Find the equation of a line parallel to \( y = -x + 4 \) that passes through \( (1, 2) \).**

#### **Solution:**
1. A parallel line has the same slope. The slope of \( y = -x + 4 \) is \( m = -1 \).
2. Using \( y = mx + b \), substitute \( m = -1 \) and \( (x, y) = (1, 2) \):
   \[
   2 = -1(1) + b
   \]
3. Solve for \( b \):
   \[
   2 = -1 + b \implies b = 3
   \]
4. The equation of the line is:
   \[
   y = -x + 3
   \]

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### Follow-Up Questions:
1. What is the difference in slope between parallel and perpendicular lines?
2. Can two lines with different slopes ever be parallel? Why or why not?
3. How can you confirm that two lines are perpendicular using their slopes?
4. What happens if the given point to go through is already on the original line?
5. How can you use graphing to verify your equations?

**Tip:** Always check your slopes carefully—parallel lines share the same slope, while perpendicular lines have slopes that multiply to \( -1 \).

Give me some problems with parallel and perpendicular lines with step-by-step answers right under (giving me a y coordinate to go through).

Explore detailed problems on parallel and perpendicular lines with step-by-step solutions. Includes finding equations of lines through specific points and verifying slopes. Perfect for Grades 8-10 students learning linear equations.

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