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Difficulty: EasySystems of Linear Equations
Consider the following system of linear equations:
2x+3y=124x+6y=24\begin{aligned} 2x + 3y &= 12 \\ 4x + 6y &= 24 \end{aligned}

Which of the following statements about this system must be true? Select all that apply.

  1. The system has infinitely many solutions.Answer
  2. The graphs of the two equations represent the exact same line in the xyxy-plane.Answer
  3. The ordered pair (3,2)(3, 2) is a solution to the system.Answer
  4. D
    The system is inconsistent and has no solution.
  5. E
    The yy-intercept of the line is (6,0)(6, 0).

Answer

The correct statements are that the system has infinitely many solutions, the graphs represent the exact same line, and (3,2)(3, 2) is a solution to the system.
Dividing 4x+6y=244x + 6y = 24 by 22 produces 2x+3y=122x + 3y = 12, showing that both equations represent the exact same line. Therefore, the system has infinitely many solutions. Substituting x=3x = 3 and y=2y = 2 yields 2(3)+3(2)=122(3) + 3(2) = 12, confirming that (3,2)(3, 2) is one of the infinitely many valid solutions.

Step-by-Step Solution

1
Analyze the relationship between the two linear equations
Dividing the second equation 4x+6y=244x + 6y = 24 by 22 gives 2x+3y=122x + 3y = 12, which is identical to the first equation.
Comparing coefficients and constants determines whether equations in a system are dependent, independent, or inconsistent.
2
Determine the number of solutions and geometric structure
Because the equations are mathematically equivalent, they describe the same line in the coordinate plane and have infinitely many intersection points.
Identical linear equations form a dependent system with infinitely many solutions.
3
Test the ordered pair (3,2)(3, 2)
Evaluating 2(3)+3(2)=6+6=122(3) + 3(2) = 6 + 6 = 12 confirms that (3,2)(3, 2) lies on the line.
Any point satisfying one equation satisfies the entire system of equivalent equations.
4
Determine the axis intercepts
Setting y=0y=0 gives 2x=12⇒x=62x=12 \Rightarrow x=6 (the xx-intercept is (6,0)(6,0)); setting x=0x=0 gives 3y=12⇒y=43y=12 \Rightarrow y=4 (the yy-intercept is (0,4)(0,4)).
Checking coordinates of axis intersections prevents mislabeling xx- and yy-intercepts.

Key Concept

Dependent Systems of Linear Equations
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Topics

Systems of Linear Equations

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