Expressions and Equations: Exponent Rules, Scientific Notation, Linear Equations, and Systems
The Expressions and Equations domain tests integer exponent rules, scientific notation, solving multi-step linear equations in one variable, the meaning of slope and intercept, and solving systems of two linear equations. A common item type asks you to decide whether an equation has one solution, no solution, or infinitely many.
Exponent rules
For nonzero a and integers m and n:
- a^m · a^n = a^(m+n)
- a^m / a^n = a^(m−n)
- (a^m)^n = a^(mn)
- (ab)^n = a^n b^n
- a^0 = 1
- a^(−n) = 1/a^n
Scientific notation
A number is in scientific notation when written a × 10^n with 1 ≤ |a| < 10. To multiply, multiply coefficients and add exponents: (3 × 10⁴)(2 × 10⁻⁶) = 6 × 10⁻². To divide, divide coefficients and subtract exponents.
Linear equations
Clear parentheses, combine like terms, and isolate the variable. If the variable cancels and the remaining statement is true (5 = 5), there are infinitely many solutions; if it is false (3 = 7), there is no solution.
In y = mx + b, m is the slope and b is the y-intercept. Slope from two points: m = (y₂ − y₁)/(x₂ − x₁).
Systems of equations
The solution is the ordered pair satisfying both equations - the intersection point on a graph. Use substitution when a variable is isolated; use elimination when coefficients are opposites or easy multiples. Parallel lines produce no solution; identical lines produce infinitely many.
Worked example
Solve 2x + 3y = 12 and x − y = 1. From the second equation, x = y + 1. Substitute into the first: 2(y + 1) + 3y = 12, so 2y + 2 + 3y = 12, 5y = 10, y = 2. Then x = 3. Check: 2(3) + 3(2) = 12 and 3 − 2 = 1, so (3, 2) is correct.
Traps
- Evaluating −3² as 9. Exponentiation binds tighter than the negative sign: −3² = −9, while (−3)² = 9.
- Distributing a negative sign to only the first term inside parentheses.
- Reporting 'no solution' when the equation produces x = 0; zero is a valid solution.
- Subtracting coordinates in inconsistent order when computing slope.
- Leaving a coefficient such as 12.5 × 10³ instead of rewriting it as 1.25 × 10⁴.