Algebra, oriented

When you substitute a value, put it in parentheses, especially when it is negative: for x = −3, x² is (−3)² = 9, not −9. Then follow the order of operations: grouping, exponents, multiplication and division from left to right, and addition and subtraction from left to right.

Combine only like terms, the ones with the same variable raised to the same power. 7x + 2x² − 3x simplifies to 2x² + 4x; the x and x² terms stay separate.

Distribute a factor to every term inside the parentheses and carry the sign: −2(3x − 5) = −6x + 10. Factoring reverses distribution by pulling out the largest factor every term shares: 8x² − 12x = 4x(2x − 3).

Rule: put every substituted value in parentheses, and when you distribute a negative, change the sign of every term inside.

To solve a linear equation, undo what was done to the variable in reverse order. In 3(x + 4) = 2x + 19, distribute (3x + 12 = 2x + 19), move the x terms to one side (x + 12 = 19), and finish (x = 7). Whatever you do, do it to both sides and to every term. Fractions are easier to clear than to carry, so multiply every term by the least common denominator.

Inequalities use the same steps with one exception: multiplying or dividing by a negative number reverses the sign. From 5 − 3x ≥ 20, subtract 5 to get −3x ≥ 15, then divide by −3 to get x ≤ −5. Test a value to confirm: x = −6 gives 5 + 18 = 23, which is at least 20.

Check every solution in the original equation, not in a later line. For x = 7 above, both sides equal 33. A check that fails usually points to a sign error or a term that was not carried to both sides.

Rule: do the same thing to both sides, flip an inequality only when multiplying or dividing by a negative, and check in the original equation.

Elimination works when one variable can be made to cancel. From 3x + 4y = 39 and 3x − y = 9, subtracting the second equation from the first gives 5y = 30, so y = 6 and then x = 5. If no coefficients match, multiply one equation first so that they do.

Substitution works when a variable is already on its own. With x = 3y − 2 and 2x + y = 17, replace x to get 2(3y − 2) + y = 17, so 7y = 21, y = 3, and x = 7. Keep the substituted expression in parentheses so the multiplication reaches every term.

When you know the sum and the difference of two numbers, the larger is half of the sum plus the difference and the smaller is half of the sum minus it: a sum of 44 and a difference of 10 give 27 and 17.

Rule: choose the method that makes a variable disappear in one step, then substitute back into an original equation to find the other.

f(5) is the output when 5 goes in. For a composition such as f(g(x)), evaluate the inner function first. With f(x) = x² + 1 and g(x) = 2x − 1, g(3) = 5 and f(5) = 26, while g(f(3)) = g(10) = 19, so the order matters.

An inverse undoes a function by reversing its steps. If f(x) = 5x − 2, f multiplies by 5 and subtracts 2, so its inverse adds 2 and divides by 5: the inverse at 43 is 9, and f(9) = 43 confirms it.

An arithmetic sequence adds a fixed difference, so its nth term is the first term plus n − 1 differences: starting at 13 with a difference of 7, the 20th term is 13 + 19 × 7 = 146. A geometric sequence multiplies by a fixed ratio: starting at 5 with a ratio of 4, the 5th term is 5 × 4⁴ = 1,280. Counting n steps instead of n − 1 is the classic off-by-one error.

Rule: work from the inside out, reverse the steps to invert, and count n − 1 steps from the first term.

Every exponent rule counts factors. Multiplying powers of one base adds the exponents, because 6⁴ × 6³ has seven factors of 6, so it equals 6⁷. A power of a power multiplies them: (2³)⁴ = 2^12. Dividing subtracts them: 5⁷ ÷ 5⁵ = 5² = 25. Multiplying the exponents when you should add them is the standard slip.

To solve an equation with the variable in an exponent, write both sides with the same base: 9^x = 27 becomes 3^(2x) = 3³, so 2x = 3 and x = 1.5.

Growth by a fixed factor is a power. An investment that doubles every 7 years grows to 2⁴ = 16 times its size in 28 years. Adding the same amount each period instead treats the growth as linear and underestimates it.

Rule: count the factors, match the bases, and treat repeated multiplication as a power.

To factor x² + bx + c, find two numbers that multiply to c and add to b. For x² + x − 30 they are 6 and −5, so the roots are −6 and 5. When factoring is hard, the quadratic formula always works: x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant, b² − 4ac, tells you in advance whether there are two, one, or no real roots.

Without solving, the roots of x² + bx + c add to −b and multiply to c, so for x² + x − 30 the sum is −1 and the product −30. Getting the sign of the sum wrong is the usual trap.

Identities save work. A difference of squares factors as (a + b)(a − b), so 81² − 79² = 160 × 2 = 320. And x² + y² = (x + y)² − 2xy, so if x + y = 13 and xy = 40, then x² + y² = 169 − 80 = 89. The remainder when a polynomial is divided by (x − a) is simply its value at a.

Rule: look for a factoring or an identity first, fall back on the formula, and watch the sign of every root.

Start by naming each unknown with its units, then translate one sentence at a time. Watch phrases that reverse order: 8 less than a number is n − 8, not 8 − n.

Some facts carry the whole problem. Two people's age difference never changes: if a mother is 34 and her son 4, she will be three times his age when 34 + t = 3(4 + t), which is in 11 years, when she is 45 and he is 15. Work rates add: one crew builds a deck in 10 hours and another in 40, so together they do 1/10 + 1/40 = 1/8 of it per hour and finish in 8 hours. A mixture of two items gives a count equation and a value equation: 30 tickets at $8 and $5 totaling $195 means a + c = 30 and 8a + 5c = 195, so 3a = 45, with 15 adult and 15 child tickets.

Finally, check that the answer fits the situation: whole numbers of people and tickets, positive times, and a result that satisfies every sentence of the problem.

Rule: name the unknowns, translate one sentence at a time, and use the structural fact the story is built on.