By Luc Braybury
Updated Aug 30, 2022
Elementary algebra forms the backbone of mathematical reasoning, allowing us to describe relationships with variables and manipulate equations that include them. Mastering the art of isolating the unknown—whether it’s a simple linear term or a complex exponential—empowers you to solve a wide range of problems efficiently and accurately.
Move all constants to the opposite side of the equation. For instance, with 4x2 + 9 = 16, subtract 9 from both sides to obtain 4x2 = 7.
Divide each side by the coefficient of the variable. From 4x2 = 7, divide by 4 to get x2 = 1.75.
Take the appropriate root to remove the exponent. From x2 = 1.75, the square root yields x ≈ 1.32.
Subtract or add constants to isolate the radical. For √(x + 27) + 11 = 15, subtract 11 to obtain √(x + 27) = 4.
Square both sides to eliminate the square root: (√(x + 27))2 = 42 ⇒ x + 27 = 16.
Isolate x by subtracting 27: x = 16 – 27 = –11.
Set the quadratic equal to zero. From 2x2 – x = 1, subtract 1 to get 2x2 – x – 1 = 0.
Factor the left‑hand side when possible. The example factors as (2x + 1)(x – 1) = 0.
Set each factor to zero and solve: 2x + 1 = 0 ⇒ x = –½ and x – 1 = 0 ⇒ x = 1.
Rewrite denominators in factored form: 1/(x – 3) + 1/(x + 3) = 10/(x2 – 9) becomes 1/(x – 3) + 1/(x + 3) = 10/((x – 3)(x + 3)).
Multiply every term by (x – 3)(x + 3) to clear denominators, resulting in (x + 3) + (x – 3) = 10.
xCombine like terms: 2x = 10 ⇒ x = 5.
Remove constants from the side containing the exponential. From 100·(14x) + 6 = 10, subtract 6 to get 100·(14x) = 4.
Divide by 100: 14x = 0.04.
Take ln of both sides: ln(14x) = ln(0.04) leading to x·ln(14) = ln(1/25).
xDivide both sides by ln(14): x = –ln(25)/ln(14) ≈ –1.22.
From 2·ln(3x) = 4, divide by 2 to get ln(3x) = 2.
Exponentiate both sides: eln(3x) = e2, simplifying to 3x = e2.
xDivide by 3: x = e2/3 ≈ 2.46.
