System of Equations on the SAT Exam

Overview

• Focus on understanding what it means for a system of equations to have:
  • No solution
  • Infinite number of solutions

Solutions in a System of Equations

• Solution: Intersection point of two lines (same x and y values).
• No Solution: Lines never intersect, meaning they are parallel (equal slopes).
• Infinite Solutions: Always intersect, meaning they are the same line (equal slopes).

Key Concepts

• For both no solution and infinite solutions, the slopes of the lines must be the same in a linear system.

Examples and Problem Solving

Example 1

• Problem: If an equation has an infinite number of solutions, find the value of m.
• Solution: Set slopes equal to each other:
  • First equation: 2mx + 8m - 3
  • Second equation: 6x + 21
  • Set 2m = 6, thus m = 3.

Example 2

• Problem: If a system has an infinite number of solutions, find the value of a/b.
• Solution:
  • Rearrange equations to slope-intercept form:
    • First equation: y = -2/5x + 12
    • Second equation: y = -a/bx + 20/b
  • Set slopes equal: -2/5 = -a/b, solving gives a/b = 2/5.
  • Shortcut: Use the ratio of coefficients when equations are lined up.

Example 3

• Problem: Given ax² + 3ax - 2x² = bx, find b.
• Solution:
  • Balance x² terms on both sides: a = 2.
  • Substitute to find b: b = 6.

Example 4

• Problem: If a system has no solution, find the value of k.
• Solution:
  • Set ratios of coefficients equal: 3/-4 = 4/-k, solve to find k = 16/3.

Example 5

• Problem: In a system with infinitely many solutions, what must equal c?
• Solution:
  • Distribute and compare: 3x + 3a = bx + c, b = 3, c = 3a.

Example 6

• Problem: If a system has no solution, find b.
• Solution:
  • Ensure same slope: left slope 5 = right slope b, thus b = 5.

Example 7

• Problem: Value of 35x + 14y given equations.
• Solution:
  • Factor: 7(5x + 2y), solve for 5x + 2y = 8, result is 56.

Example 8

• Problem: Solve for x - y.
• Solution:
  • Use elimination method to solve system:
    • Eliminate y by multiplying and adding equations.
    • x = 1, y = -2, thus x - y = 3.

Conclusion

• Understanding systems of equations, especially concepts of no solution and infinite solutions, is crucial for the SAT.
• Practice different methods like elimination and shortcuts for efficiency.