Mathematics · 9 min read
Algebraic Thinking
Variables and Expressions
A variable is a letter or symbol that stands for an unknown or changing quantity. An algebraic expression combines numbers, variables, and operations (3x + 5, 2a − 7b, x²/4) but has no equal sign. An equation states that two expressions are equal (3x + 5 = 20), and an inequality compares them with <, >, ≤, ≥, or ≠.
Evaluating and Simplifying Expressions
To evaluate an expression, substitute the given values for the variables and follow the order of operations. To simplify, apply the distributive property and combine like terms; the value of the expression does not change.
- Evaluate 2x² − 3y for x = 4 and y = 5: 2(4)² − 3(5) = 2(16) − 15 = 32 − 15 = 17
- Simplify 3(2x + 4) − 5x + 7 = 6x + 12 − 5x + 7 = x + 19
- Simplify 4(a − 3) − 2(a + 1) = 4a − 12 − 2a − 2 = 2a − 14
Translating Words into Algebra
| Operation | Key words | Example → algebra |
|---|---|---|
| Addition | sum, total, more than, increased by, plus | a number increased by 6 → n + 6 |
| Subtraction | difference, less than, fewer than, decreased by, minus | 5 less than a number → n − 5 |
| Multiplication | product, times, of, twice, double | twice a number → 2n |
| Division | quotient, divided by, per, ratio, half of | the quotient of a number and 4 → n/4 |
| Equals | is, equals, gives, results in, is the same as | three more than twice a number is 17 → 2n + 3 = 17 |
Solving Equations
To solve an equation, find the value of the variable that makes it true. Use inverse operations to undo what has been done to the variable, doing the same thing to both sides to keep the equation balanced. Undo addition/subtraction first, then multiplication/division. Check by substituting the solution back into the original equation.
- One step: x + 5 = 12 → x = 7 (subtract 5); 3x = 15 → x = 5 (divide by 3)
- Two steps: 2x + 3 = 11 → 2x = 8 → x = 4
- Multi-step: 3(x − 2) + 4 = 19 → 3x − 6 + 4 = 19 → 3x − 2 = 19 → 3x = 21 → x = 7
- Variables on both sides: 5x + 3 = 2x + 15 → 3x + 3 = 15 → 3x = 12 → x = 4
- Check: 5(4) + 3 = 23 and 2(4) + 15 = 23 ✓
Formulas and Literal Equations
A formula is an equation that relates quantities, such as distance d = rt, perimeter of a rectangle P = 2l + 2w, simple interest I = Prt, and temperature F = (9/5)C + 32. Substitute the known values and solve for the unknown, or rearrange the formula first (solving d = rt for t gives t = d/r).
- A car travels 150 miles in 2.5 hours. r = d ÷ t = 150 ÷ 2.5 = 60 miles per hour
- $500 is invested at 4% simple interest for 3 years. I = Prt = 500 × 0.04 × 3 = $60
- A rectangle has perimeter 30 cm and length 9 cm. 30 = 2(9) + 2w → 12 = 2w → w = 6 cm
Inequalities
Solve an inequality the same way as an equation, with one exception: multiplying or dividing both sides by a negative number reverses the inequality sign (−3x > 12 → x < −4). Graph the solution on a number line with an open circle for < or > (the endpoint is not included) and a closed circle for ≤ or ≥ (the endpoint is included), then shade the direction of the solutions.
Number and Shape Patterns
A pattern follows a rule. In a repeating pattern a core unit repeats (red, blue, green, red, blue, green, …). To find a far term, divide the position by the length of the core and use the remainder: the 50th item has 50 ÷ 3 = 16 R2, so it is the 2nd item in the core — blue. (A remainder of 0 means the last item of the core.)
In a growing pattern each figure or term changes by a rule. Make a table of figure number and quantity, find how much it changes each time, and write a rule that works for any figure number n.
Arithmetic sequences add a constant difference d: aₙ = a₁ + (n − 1)d. For 2, 5, 8, 11, … (d = 3), the 10th term is 2 + 9(3) = 29. Geometric sequences multiply by a constant ratio r: aₙ = a₁ × rⁿ⁻¹. For 3, 6, 12, 24, … (r = 2), the 5th term is 3 × 2⁴ = 48. Decide whether a pattern is additive (same difference) or multiplicative (same ratio) before extending it.
Functions and Linear Relationships
A function assigns exactly one output to each input. In function notation, f(x) = 2x + 3 means “double the input, then add 3”, so f(4) = 2(4) + 3 = 11. The domain is the set of possible inputs and the range is the set of possible outputs. An input-output table shows the rule numerically.
| x (input) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y (output) | 5 | 8 | 11 | 14 | 17 |
In this table y increases by 3 each time x increases by 1, so the rate of change (slope) is 3, and the value when x = 0 is 5 (the y-intercept or starting value): y = 3x + 5. A linear function has a constant rate of change and graphs as a straight line; its slope-intercept form is y = mx + b (often written f(x) = mx + b). A relationship is proportional only if it has the form y = kx: the ratio y/x is the same for every pair and the graph passes through the origin. y = 3x + 5 is linear but not proportional.
Coordinate Plane and Graphing
The coordinate plane is formed by a horizontal x-axis and a vertical y-axis that meet at the origin (0, 0). An ordered pair (x, y) gives horizontal position first, then vertical position. The axes divide the plane into four quadrants, numbered counterclockwise starting at the upper right.
To graph a linear equation, make a table of values, plot the points, and draw the line through them — or plot the y-intercept b and use the slope m (rise over run) to find more points.
Slope
Slope measures steepness and direction: slope = rise/run = (y₂ − y₁)/(x₂ − x₁). For the points (1, 2) and (4, 11): slope = (11 − 2)/(4 − 1) = 9/3 = 3. In a real-world graph, slope is a rate (dollars per hour, miles per gallon), and the y-intercept is the starting amount.
Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals (2 and −1/2), so their product is −1.
Systems of Equations
A system of equations is two or more equations with the same variables; its solution makes every equation true at once. Solve by graphing (the intersection point), substitution (solve one equation for a variable and substitute into the other), or elimination (add or subtract the equations to eliminate a variable).
Word Problems and Algebraic Modeling
Most algebra items on the exam are set in a real-world context. Use a consistent process:
- Read carefully and identify what is being asked.
- Define a variable for the unknown quantity.
- Write an equation or inequality that represents the relationships.
- Solve it.
- Check the solution in the context of the problem (is it reasonable? whole number needed?).
- State the answer with appropriate units.
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