Study guide: Linear and Quadratic Equations

📋 Course Outline

1. Fundamental Equation Concepts
2. Equation Classification
3. Linear Equations with n Unknowns
4. Automotive Services System

📖 1. Fundamental Equation Concepts

🔑 Key Concepts & Definitions

• Equation: An equality containing one or more unknowns, represented by letters for unknown values.
• Solution of an equation: The value or values that make the equality true when substituted into the equation.
• Degree of an equation: The greatest exponent to which the unknown is raised.

📝 Essential Points

• For the first-degree equation $3x-5=10$, solving gives $3x=15$ and $x=5$.

• For the second-degree equation $x^2-4x+3=0$, factorization gives $(x-1)(x-3)=0$ and the solutions $x=1$ and $x=3$.

💡 Memory Hook

Unknown → equality → solution → degree

📖 2. Equation Classification

★ Must-know

📌 First-degree or linear equations have the unknown raised to exponent 1, such as $2x+1=5$, whereas second-degree or quadratic equations have the unknown raised to exponent 2, such as $x^2-3x+2=0$.

• By the number of unknowns, an equation may have:

  • one unknown
  • two unknowns
  • three or more unknowns

• By the number of solutions, equations may be:

  • determined: one or more real solutions
  • indeterminate: infinitely many solutions
  • incompatible: no solution

Further detail

• Higher-degree equations have exponents greater than 2, as in $x^3-x^4+8=0$.

• The system $x+y=6$ and $x-y=2$ is compatible determined and has the solution $x=4, y=2$.

💡 Memory Hook

Determined has real solutions, indeterminate has infinitely many, incompatible has none

📖 3. Linear Equations with n Unknowns

★ Must-know

📐 Formula - A linear equation with n unknowns has the form $a_1x_1+a_2x_2+\cdots+a_nx_n=b$, where the coefficients $a_1,a_2,\ldots,a_n$ and the constant $b$ are known real numbers.

📐 Formula - In the automotive example, the service counts satisfy $x+y+z=20$, $1000x+500y+750z=15000$, and $x=2z$.

Further detail

• Linear equations with n unknowns are generally solved as part of a linear system containing several equations.

📖 4. Automotive Services System

★ Must-know

• Substituting $x=2z$ into the first two equations produces $y+3z=20$ and $500y+2750z=15000$.

• Replacing $y$ with $20-3z$ gives $500(20-3z)+2750z=15000$, which simplifies to $1250z=5000$ and then $z=4$.

• Using $z=4$ gives $y=20-3(4)=8$ and $x=2(4)=8$, so the workshop performed 8 tune-ups, 8 oil changes, and 4 brake inspections.

Further detail

• The automotive system represents tune-ups by x, oil changes by y, and brake inspections by z, with prices of $1,000, $500, and $750 respectively.

💡 Memory Hook

Substitute x=2z → eliminate y → find z → recover y and x