This page contains linear equations with variables on both sides, focusing on solving for $x$. Let’s solve the equations systematically.

Task 1: Solve the given equations (Section 1)

a) $20x - 32 = 11x + 13$

1. Move $11x$ to the left:
   $20x - 11x - 32 = 13$
   $9x - 32 = 13$

2. Add 32 to both sides:
   $9x = 45$

3. Divide by 9:
   $x = 5$

b) $13x - 43 = 7x - 19$

1. Move $7x$ to the left:
   $13x - 7x - 43 = -19$
   $6x - 43 = -19$

2. Add 43 to both sides:
   $6x = 24$

3. Divide by 6:
   $x = 4$

c) $5x + 21 = 101 - 11x$

1. Move $11x$ to the left:
   $5x + 11x + 21 = 101$
   $16x + 21 = 101$

2. Subtract 21 from both sides:
   $16x = 80$

3. Divide by 16:
   $x = 5$

d) $15 + 9x = 90 - 16x$

1. Move $16x$ to the left:
   $15 + 9x + 16x = 90$
   $15 + 25x = 90$

2. Subtract 15 from both sides:
   $25x = 75$

3. Divide by 25:
   $x = 3$

e) $18x - 63 = 5x - 24$

1. Move $5x$ to the left:
   $18x - 5x - 63 = -24$
   $13x - 63 = -24$

2. Add 63 to both sides:
   $13x = 39$

3. Divide by 13:
   $x = 3$

Task 2: Solve for $x$ (Section 2)

a) $-8x - 45 = -6x - 13$

1. Move $-6x$ to the left:
   $-8x + 6x - 45 = -13$
   $-2x - 45 = -13$

2. Add 45 to both sides:
   $-2x = 32$

3. Divide by $-2$:
   $x = -16$

b) $65 + 9x = 20 + 24x$

1. Move $24x$ to the left:
   $65 + 9x - 24x = 20$
   $65 - 15x = 20$

2. Subtract 65 from both sides:
   $-15x = -45$

3. Divide by $-15$:
   $x = 3$

c) $-27 + 5x = -3x - 67$

1. Move $-3x$ to the left:
   $-27 + 5x + 3x = -67$
   $-27 + 8x = -67$

2. Add 27 to both sides:
   $8x = -40$

3. Divide by 8:
   $x = -5$

Let me know if you would like detailed explanations for any specific equation!

Questions to expand:

1. What is the process for identifying the operations needed to isolate $x$?
2. How do we handle negative coefficients when solving equations?
3. Can these techniques be applied to quadratic equations? If not, why?
4. What happens when the variable cancels out, and no solution or infinite solutions arise?
5. How can these equations be verified by substitution?

Tip:

Always double-check your solutions by substituting them back into the original equations!