Solved Problems: Infinitely Many Solutions in Linear Systems (Algebra 2 Examples)

Question

Hey everyone! 👋 Let's tackle linear systems with infinitely many solutions. It can be a bit tricky, but I'll break it down for you. We'll go through a quick study guide and then test your knowledge with a quiz! Good luck! 🍀

danielle324 · Accepted Answer

📚 Quick Study Guide

♾️ A linear system has infinitely many solutions when the equations represent the same line.
  📝 This occurs when, after simplifying, the equations are multiples of each other.
  ➗ You can identify this by checking if one equation can be obtained by multiplying the other equation by a constant.
  📐 In slope-intercept form ($y = mx + b$), both equations will have the same slope ($m$) and y-intercept ($b$).
  💡 When solving using substitution or elimination, you'll end up with an identity like $0 = 0$.

Practice Quiz

Question 1: Which of the following systems of equations has infinitely many solutions?
    
      $x + y = 5$, $2x + 2y = 8$
      $x - y = 3$, $2x - 2y = 6$
      $x + y = 2$, $x - y = 2$
      $y = 3x + 1$, $y = -3x + 1$

Question 2: What condition must be met for a linear system to have infinitely many solutions?
    
      The lines are parallel.
      The lines intersect at one point.
      The lines are the same.
      The lines are perpendicular.

Question 3: Which system results in an identity (e.g., $0 = 0$) when solved using elimination?
    
      $2x + y = 4$, $4x + 2y = 6$
      $3x - y = 1$, $6x - 2y = 2$
      $x + y = 7$, $x - y = 1$
      $5x + 5y = 10$, $x + y = 3$

Question 4: Identify the system where one equation is a scalar multiple of the other.
    
      $y = x + 1$, $y = x + 2$
      $y = 2x$, $2y = 4x$
      $y = x$, $x + y = 1$
      $y = -x + 3$, $y = x - 3$

Question 5: Given the equation $y = 2x + 3$, which of the following equations, when paired with it, will result in infinitely many solutions?
    
      $y = -2x + 3$
      $2y = 4x + 6$
      $y = 2x - 3$
      $y = 3x + 2$

Question 6: What does it mean graphically when a system of linear equations has infinitely many solutions?
    
      The lines are skewed.
      The lines are overlapping.
      The lines are diverging.
      The lines are intersecting at multiple points but not overlapping completely.

Question 7: Which of the following systems has infinitely many solutions?
    
      $x + y = 1$, $x - y = 1$
      $2x + 2y = 4$, $x + y = 2$
      $x = 3$, $y = 4$
      $y = x + 5$, $y = x - 5$

Click to see Answers

B
  C
  B
  B
  B
  B
  B

Solved Problems: Infinitely Many Solutions in Linear Systems (Algebra 2 Examples)

1 Answers

📚 Quick Study Guide

Practice Quiz

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