Algebra 1 Formulas with Worked Equation Examples

Start with a linear equation

For ax + b = c with a nonzero, x = (c − b)/a. In 5x − 7 = 18, adding 7 gives 5x = 25 and dividing by 5 gives x = 5. Check 5×5 − 7 = 18. The nonzero condition matters: if a = 0, the equation instead becomes a statement about b and c and may be either always true or inconsistent. A formula sheet should include such conditions beside the expression.

Use the line formula with its definitions

A nonvertical line has form y = mx + b, where m is slope and b is the y-intercept. For points (1,4) and (3,10), m = (10 − 4)/(3 − 1) = 3. Substituting the first point gives 4 = 3×1 + b, so b = 1 and y = 3x + 1. This constructs a two-variable relationship. To use the one-variable equation tool, ask a specific question such as 3x + 1 = 16, which gives x = 5.

Apply exponent rules to matching bases

For a nonzero base a, a^m × a^n = a^(m+n) and a^m/a^n = a^(m−n). For example, 2³×2⁴ = 2⁷ = 128. These rules do not turn 2³ + 2⁴ into 2⁷; the sum is 8 + 16 = 24. Parentheses also distinguish (−3)² = 9 from −(3²) = −9. State whether an operation is addition, multiplication or exponentiation before selecting a familiar-looking rule.

Use factoring when a product equals zero

The quadratic x² − 5x + 6 = 0 factors as (x − 2)(x − 3) = 0, giving x = 2 or x = 3. Substitution verifies both. The zero-product rule applies because the product equals zero, not merely because two factors appear. For a general quadratic ax² + bx + c = 0 with a nonzero, the quadratic formula supplies the roots when simple factoring is inconvenient. Keep both roots until the original problem supplies a reason to exclude one.

Keep expressions, equations and solutions distinct

An expression such as 3x + 2 describes a value that depends on x. An equation such as 3x + 2 = 14 asserts equality and can be solved for values that make it true. Simplifying an expression does not automatically find a solution because there may be no equality to satisfy. Likewise, a formula containing several variables expresses a relationship rather than supplying all their values. Identify whether the question asks you to simplify, evaluate, rearrange, solve or construct a model before selecting the calculator input.

Preserve equivalence and check the original statement

When solving an equation, apply the same permitted operation to both sides. Distribute a multiplier to every term inside its parentheses, preserve negative signs, and do not divide by an expression that might be zero without treating that case separately. Substitute each candidate into the original equation, not only the last transformed version. Squaring, cancelling or multiplying by a variable denominator can alter the set of allowed values. An answer that satisfies a rearranged equation can still violate the original domain or the physical meaning of a length or count.

Match the local calculator to its supported task

The local equation forms solve linear and quadratic polynomial equations in one variable named x. They are not general symbolic algebra systems, multi-variable formula rearrangers or inequality solvers. Use explicit parentheses for grouped numerators and denominators and use an equals sign when entering an equation so the intended sides are clear. For a geometric or applied problem, derive the equation first and keep the relevant units and constraints in the written working. A numerical solution does not by itself explain which original measurement or variable that number represents.

Rearrange a formula for its intended unknown

For a rectangle formula A = lw, solving for width gives w = A/l when l is nonzero. With area 42 and length 7, width is 6. This is a formula rearrangement followed by substitution, rather than a claim that the single-variable equation form accepts the letters A, l and w simultaneously. Keep the distinction visible when moving between a formula sheet and a numeric input form.

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