Algebra Didn’t Have to Be This Hard

Algebra is arithmetic with a blank in it: give the missing number a name, write down how the numbers fit together, work back to it by undoing the steps in reverse order, and check by putting the answer back in.

A letter stands for a number: an unknown is one hidden number, a variable can take many values. An expression is a recipe for a number and an equation says two amounts are equal; to evaluate an expression, substitute each letter's number, writing the multiplication signs back in.

So that a written calculation means one thing to everyone, we agree on an order: brackets, then powers, then multiplication and division from left to right, then addition and subtraction from left to right. A fraction bar is hidden brackets, and when a reader could hesitate, add brackets.

Negative numbers sit to the left of zero on the number line. Adding a negative moves left, subtracting a negative moves right (it's the same as adding), and when multiplying or dividing, same signs give a positive and different signs a negative; each rule is what keeps arithmetic working on both sides of zero.

To turn words into algebra, name the letter with its units, translate phrase by phrase ("less than" reverses the order), and test the result with one number you can check in your head. A table with a steady jump is a straight-line rule: the jump times n, plus the value at zero.

An equation is a level balance, so you can do any move you like as long as you do it to both sides; choose the undo (inverse operation) of whatever was done to the letter until it stands alone, then put the answer back into the original equation to check.

When several things have been done to the letter, undo them in reverse order, last on, first off; when the letter is on both sides, gather it on one side first; and when the letters vanish, the leftover statement tells you there is no solution (if it's false) or that every number works (if it's always true).

Like terms have the same letter part and collect by adding their numbers; a number outside a bracket multiplies everything inside it (the distributive law, an area picture), and factoring pulls a shared factor back out; none of this changes the value, so test any tidying with an ordinary number like 7.

Clear fractions by multiplying every term on both sides by a common denominator, and decimals by multiplying by ten or a hundred; every percent problem is part = rate × whole, and a percent change is undone by dividing by its multiplier, never by adding or subtracting the percent.

A formula is an equation with several letters, and you can solve it for any one of them with the same balance moves, treating the other letters as numbers; rearrange first, then put in the numbers, and check by putting the answer back into the original formula.

A coordinate grid turns a pair of numbers into a place (across first, then up) and a rule into a picture; a straight row of points means equal steps, and solving an equation is reading the picture backward.

Slope is rise over run, the change up for each step across; it's every "per" in real life (miles per hour, dollars per sheet), positive going up, negative going down, zero when flat.

Every straight line except a vertical one is a start plus a rate, y = mx + b; two points pin down both numbers, and where two lines cross is where two deals cost the same.

Two unknowns need two different facts; the answer is the one pair that fits both, where their lines cross, found by graphing, substitution or elimination.

An inequality is an equation with room in it: solve it with the same balance moves, flip the sign only when multiplying or dividing by a negative, and let the story decide how to round.

An exponent counts how many copies of the base are multiplied, so multiplying powers adds the counts, dividing subtracts them, a power of a power multiplies them, a zero exponent leaves 1, and a negative exponent divides.

A square root undoes a square, an equation like x² = 49 has two answers of which the story keeps the one that fits, and the Pythagorean theorem a² + b² = c² turns any right triangle into a square-root problem.

A quadratic has an x² term, its graph is a parabola, and its solutions are where the curve crosses zero; factoring with the zero-product rule, completing the square or the quadratic formula finds them, and the vertex marks the best or worst case.

Linear growth adds the same amount each step and draws a straight line; exponential growth multiplies by the same factor, starts slowly, doubles at a steady pace and always overtakes the line in the end, so ask "over how long?" before comparing.

A function is a machine that turns each input into exactly one output; f(x) names the output, a table with equal steps reveals a line's rule, the inverse runs the machine backward (which is solving an equation), and the domain says which inputs make sense.

A word problem is a story told twice, once in words and once in algebra. Picture it, name one unknown with its unit, write the one thing the story says is equal, solve, and check the answer against the words, not just the equation.

A spreadsheet is algebra with addresses for letters: a formula is an expression, dragging it down is substitution, dollar signs turn a letter into a constant, and goal seek is solving an equation. Read every formula aloud with units, and test one row by hand.

In direct proportion the ratio stays fixed (y = kx: double one, double the other); in inverse proportion the product stays fixed (xy = k: double one, halve the other); anything with a fixed fee or starting value is neither. Test a table by dividing pairs and by multiplying them.

Algebra is ten habits used again and again: name the unknown, picture it, keep the balance, undo in reverse order, tidy before solving, clear the fractions, check by putting it back, draw the graph, watch units and signs, and ask whether the answer makes sense.