[Solved] Linear equations

Assignment Prompt & Instructions

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Here are some challenging linear equation problems, ranging from tricky algebra to systems with special conditions:

1. Fractional coefficients

Solve for x:

(2x – 1)/3 – (x + 4)/5 = (3x – 2)/15 + 1

2. Variables on both sides with nested brackets

Solve for x:

5[3 – 2(x – 4)] = 3[2(x + 1) – 5] – 7

3. Literal equation (solve for a variable in terms of others)

Solve for x:

a(x – b) + b(x – a) = 2ab

4. Equation with a parameter — find the value that makes it have no solution

For what value of k does the equation have no solution?

3(x – 2) + k = 2x – (5 – x)

5. System of three equations, three unknowns

Solve:

x + y + z = 6

2x – y + z = 3

x + 2y – z = 5

6. System with a parameter — find k so the system has infinitely many solutions

kx + 2y = 6

3x + (k – 1)y = 9

7. Equation with variables in denominators (reduces to linear)

Solve for x:

1/(x – 2) + 1/(x + 2) = 4/(x² – 4)

8. Word-problem style linear system (age problem)

A father's age is 3 times his son's age. In 12 years, the father will be twice as old as his son. Find both present ages — then check whether the equations are truly linear or need care in setup.

9. Absolute value linear equation

Solve for x:

|2x – 5| + 3 = |x + 1| + 6

10. Consistency check — determine if the system has one solution, no solution, or infinite solutions

2x – 3y = 7

-4x + 6y = -14