https://file.aiquickdraw.com/mathbot/file/17253367295441guh6vb7.pdf

hello! please just show the steps (not in words) for each of the equation and make it as a highschooler

Below are the steps for each equation as requested:

### **Finding the Equation of a Line:**
1. **(1) Through points (0, 1) and (1, 2):**
   - Slope: \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 1}{1 - 0} = 1 \)
   - Point-slope form: \( y - y_1 = m(x - x_1) \)
   - Substitute: \( y - 1 = 1(x - 0) \)
   - Equation: \( y - 1 = x \) or \( y = x + 1 \)

2. **(2) Through points (-3, -1) and (5, -2):**
   - Slope: \( m = \frac{-2 + 1}{5 + 3} = -\frac{1}{8} \)
   - Point-slope form: \( y - y_1 = m(x - x_1) \)
   - Substitute: \( y + 1 = -\frac{1}{8}(x + 3) \)
   - Equation: \( y + 1 = -\frac{1}{8}x - \frac{3}{8} \) or \( y = -\frac{1}{8}x - \frac{11}{8} \)

3. **(3) Through (4, 4) and parallel to \( y = \frac{9}{4}x \):**
   - Slope: \( m = \frac{9}{4} \)
   - Point-slope form: \( y - 4 = \frac{9}{4}(x - 4) \)
   - Equation: \( y - 4 = \frac{9}{4}x - 9 \) or \( y = \frac{9}{4}x - 5 \)

4. **(4) Through (3, 4) and perpendicular to \( y = -\frac{3}{5}x - 4 \):**
   - Slope: \( m = \frac{5}{3} \) (negative reciprocal)
   - Point-slope form: \( y - 4 = \frac{5}{3}(x - 3) \)
   - Equation: \( y - 4 = \frac{5}{3}x - 5 \) or \( y = \frac{5}{3}x - 1 \)

5. **(5) Through (5, 3) with slope = 0:**
   - Slope: \( m = 0 \)
   - Equation: \( y = 3 \)

### **Solving Equations for \( x \):**
1. **(6) \( 5 = 7x - 16 \)**
   - \( 5 + 16 = 7x \)
   - \( 21 = 7x \)
   - \( x = 3 \)

2. **(7) \( 2x - 3 = 5 - x \)**
   - \( 2x + x = 5 + 3 \)
   - \( 3x = 8 \)
   - \( x = \frac{8}{3} \)

3. **(8) \( \frac{1}{2}(x - 3) + x = 17 + 3(4 - x) \)**
   - \( \frac{x}{2} - \frac{3}{2} + x = 17 + 12 - 3x \)
   - \( \frac{3x}{2} - \frac{3}{2} = 29 - 3x \)
   - Multiply by 2: \( 3x - 3 = 58 - 6x \)
   - \( 9x = 61 \)
   - \( x = \frac{61}{9} \)

4. **(9) \( \frac{5}{x} = \frac{2}{x-3} \)**
   - Cross-multiply: \( 5(x - 3) = 2x \)
   - \( 5x - 15 = 2x \)
   - \( 3x = 15 \)
   - \( x = 5 \)

5. **(10) \( 2x + 4 \geq 3 \)**
   - \( 2x \geq -1 \)
   - \( x \geq -\frac{1}{2} \)

6. **(11) \( -2x + 4 \geq 3 \)**
   - \( -2x \geq -1 \)
   - \( x \leq \frac{1}{2} \)

Would you like any further details or have any questions? Here are some related questions to consider:

1. How do you find the slope of a line given two points?
2. How do you derive the point-slope form of a line?
3. What is the difference between solving linear equations and solving inequalities?
4. How do you identify when lines are parallel or perpendicular based on their slopes?
5. What are the steps to factor quadratic equations?

**Tip:** Always check your solutions by substituting them back into the original equations to verify their accuracy.

Explore step-by-step solutions to finding the equation of a line and solving various equations, including point-slope form and cross-multiplication methods. Suitable for high school students learning algebra and linear equations.

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