Question

Create a single variable linear equation that has no solution. Solve the equation algebraically to prove that it does not have a solution.
Create a single variable linear equation that has one solution. Solve the equation algebraically to prove that there is one distinct solution for the equation.
Create a single variable linear equation that has infinitely many solutions. Solve the equation algebraically to prove that there is an infinite number of solutions for the equation

WRITER

Answer 1

1) No solution
Equation: 2x + 3 = 2x − 1

Solve:
2x + 3 = 2x − 1
Subtract 2x from both sides:
3 = −1

This is a contradiction (3 ≠ −1). Since the variable cancels and the remaining statement is false, there is no value of x that satisfies the equation. Solution set: ∅.

2) One solution
Equation: 4x − 5 = 11

Solve:
4x − 5 = 11
Add 5 to both sides:
4x = 16
Divide both sides by 4:
x = 4

Because we divided by the nonzero coefficient 4, exactly one value of x satisfies the equation. Solution set: {4}.

3) Infinitely many solutions
Equation: 5(x − 2) = 5x − 10

Solve:
5(x − 2) = 5x − 10
Expand left side:
5x − 10 = 5x − 10
Subtract 5x from both sides:
−10 = −10

This is a tautology (always true). Since the variable cancels and the remaining statement is true for every x, every real number satisfies the equation. Solution set: all real numbers (infinitely many solutions).