Common Algebraic Equations

Why This Matters

In College Algebra, you're not just learning to solve equations—you're building a toolkit for modeling how the world actually works. Every equation type you encounter represents a different kind of relationship: linear equations describe constant rates of change, exponential equations capture growth and decay, and quadratic equations model situations where change itself is changing. The PSAT and college math courses expect you to recognize which equation type fits a given scenario and manipulate it confidently.

Here's the key insight: these equations aren't isolated topics. They connect through shared concepts like intercepts, transformations, and inverse relationships. When you understand why a logarithm undoes an exponential or how a system of equations finds intersection points, you're thinking like the test wants you to think. Don't just memorize formulas—know what behavior each equation type models and when to reach for it.

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Linear and Proportional Relationships

These equations model situations where change happens at a constant rate. The defining feature is that equal changes in x always produce equal changes in y.

Linear Equations

• Standard form: $y = mx + b$—where $m$ represents the slope (rate of change) and $b$ is the y-intercept
• Slope interpretation—a slope of $m = 3$ means y increases by 3 units for every 1-unit increase in x; negative slopes indicate inverse relationships
• Applications span prediction and trend analysis—anytime you see "constant rate," "per unit," or "steady increase," think linear

Systems of Linear Equations

• Multiple equations sharing variables—solutions occur where all equations are simultaneously true, representing intersection points
• Three solution types exist: unique (lines intersect once), infinite (lines overlap completely), or none (parallel lines never meet)
• Solving methods include substitution, elimination, and graphing—elimination works best when coefficients align; substitution shines when one variable is already isolated

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Polynomial and Quadratic Behavior

Polynomials involve variables raised to whole-number powers. The degree of the polynomial determines the maximum number of roots and the general shape of its graph.

Quadratic Equations

• Standard form: $ax^2 + bx + c = 0$—graphs as a parabola opening upward when $a > 0$ and downward when $a < 0$
• The discriminant $b^2 - 4ac$ reveals root nature—positive means two real roots, zero means one repeated root, negative means complex roots
• Solve using factoring, completing the square, or the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$—the vertex at $x = \frac{-b}{2a}$ gives the maximum or minimum value

Polynomial Equations

• General form involves terms like $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0$—the highest power (degree) determines end behavior and maximum turning points
• The Fundamental Theorem of Algebra guarantees n roots (counting multiplicity and complex roots) for a degree-n polynomial
• Solving strategies include factoring, synthetic division, and the Rational Root Theorem—test possible rational roots of the form $\pm\frac{p}{q}$ where p divides the constant term and q divides the leading coefficient

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Exponential and Logarithmic Relationships

These equations model multiplicative change—situations where quantities double, halve, or change by percentages. Exponentials and logarithms are inverse operations, just like multiplication and division.

Exponential Equations

• Standard form: $y = a \cdot b^x$—where $a$ is the initial value and $b$ is the growth/decay factor
• Growth occurs when $b > 1$; decay occurs when $0 < b < 1$—population doubling uses $b = 2$; half-life problems use $b = 0.5$
• Solve for unknown exponents using logarithms—if $5^x = 125$, then $x = \log_5(125) = 3$

Logarithmic Equations

• Standard form: $y = \log_a(x)$—answers the question "what power of $a$ gives $x$?"
• Key properties simplify complex expressions: product rule $\log(xy) = \log x + \log y$, quotient rule $\log(\frac{x}{y}) = \log x - \log y$, power rule $\log(x^n) = n\log x$
• Logarithmic scales compress large ranges—the Richter scale and decibel scale both use logarithms to make huge variations manageable

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Equations with Special Structures

These equation types require careful attention to domain restrictions and potential extraneous solutions. The solving process itself can introduce false answers.

Absolute Value Equations

• Standard form: $|x| = a$—represents distance from zero, yielding two solutions: $x = a$ and $x = -a$ (when $a \geq 0$)
• Graphs form a V-shape—the vertex occurs where the expression inside equals zero; the graph reflects across this point
• No solution exists when $|x| = a$ and $a < 0$—distance cannot be negative, so recognize these impossible cases immediately

Rational Equations

• Involve polynomial fractions like $\frac{p(x)}{q(x)}$—solve by finding common denominators or cross-multiplying
• Domain restrictions are critical—identify values that make the denominator zero; these create vertical asymptotes and must be excluded from solutions
• Common in rate and work problems—"working together" scenarios often produce equations like $\frac{1}{a} + \frac{1}{b} = \frac{1}{t}$

Radical Equations

• Involve roots like $\sqrt{x}$ or $\sqrt[3]{x}$—solve by isolating the radical, then raising both sides to the appropriate power
• Squaring can introduce extraneous solutions—always check answers in the original equation; some "solutions" won't actually work
• Domain considerations matter—square roots require non-negative radicands in the real number system

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Inequalities and Solution Sets

Unlike equations that find specific values, inequalities describe ranges of values that satisfy a condition. Solutions are intervals or regions, not single points.

Inequalities and Their Graphs

• Symbols include $<, >, \leq, \geq$—strict inequalities ($<, >$) use open circles or dashed lines; inclusive inequalities ($\leq, \geq$) use closed circles or solid lines
• Multiplying or dividing by negatives flips the inequality sign—this is the most common error; if you divide by $-2$, change $<$ to $>$
• Systems of inequalities create feasible regions—shade the overlap of all conditions; vertices of these regions often represent optimal solutions in applications

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Quick Reference Table

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Self-Check Questions

1. Which two equation types are inverses of each other, and how would you use one to solve for an unknown exponent in the other?

2. Both rational and radical equations can produce extraneous solutions. What causes this in each case, and what should you always do before finalizing your answer?

3. Compare the graphs of $y = x^2$ and $y = |x|$. What shape does each produce, and where does each have its minimum value?

4. A system of linear equations has no solution. What does this tell you about the graphs of the two equations, and what term describes their relationship?

5. If a PSAT problem describes a quantity that "doubles every 3 years," which equation type should you use, and what would the base of the equation represent?