Answered

Select all of the ordered pairs $(x, y)$ that are solutions to the linear equation $2 x+3 y=6$.


A$(0,2)$
B$(0,6)$
C$(2,3)$
D$(3,-2)$
E$(3,0)$

F. $(6,-2)$

Verified Answer :

  • Identify that an ordered pair is a solution to \(2x + 3y = 6\) if substituting it yields a true result.
  • Test each given pair by calculating \(2x + 3y\) for each and comparing it to 6.
  • Determine that the pairs \((0, 2)\), \((3, 0)\), and \((6, -2)\) are the ones that result in a sum of 6.
  • The solutions are \(\boxed{A, E, F}\).

Explanation

  1. Understanding the Problem and Data Analysis

We are given the linear equation \(2x + 3y = 6\) and a list of six candidate ordered pairs: \((0, 2)\), \((0, 6)\), \((2, 3)\), \((3, -2)\), \((3, 0)\), and \((6, -2)\). An ordered pair \((x, y)\) is a solution to the equation if substituting its values for \(x\) and \(y\) results in a true statement, meaning the left-hand side equals 6.

  1. Testing the Candidate Ordered Pairs

We test each candidate pair by substituting the coordinates into the expression \(2x + 3y\):

  • For \((0, 2)\): \(2(0) + 3(2) = 0 + 6 = 6\). Since the result is 6, \((0, 2)\) is a solution.
  • For \((0, 6)\): \(2(0) + 3(6) = 0 + 18 = 18\). Since \(18 eq 6\), \((0, 6)\) is not a solution.
  • For \((2, 3)\): \(2(2) + 3(3) = 4 + 9 = 13\). Since \(13 eq 6\), \((2, 3)\) is not a solution.
  • For \((3, -2)\): \(2(3) + 3(-2) = 6 - 6 = 0\). Since \(0 eq 6\), \((3, -2)\) is not a solution.
  • For \((3, 0)\): \(2(3) + 3(0) = 6 + 0 = 6\). Since the result is 6, \((3, 0)\) is a solution.
  • For \((6, -2)\): \(2(6) + 3(-2) = 12 - 6 = 6\). Since the result is 6, \((6, -2)\) is a solution.
  1. Conclusion

Based on our calculations, the ordered pairs that satisfy the equation \(2x + 3y = 6\) are \((0, 2)\), \((3, 0)\), and \((6, -2)\). These correspond to the options A, E, and F.

Examples

In real life, linear equations are essential for balancing budgets or planning logistics. Imagine you have a budget of \(\$6\) to buy items where one type costs \(\$2\) and another costs \(\$3\). The equation \(2x + 3y = 6\) helps you determine all possible combinations \((x, y)\) of items you can buy without exceeding or falling short of your budget. For example, buying 3 items of the first type and 0 of the second, or 0 of the first and 2 of the second, are both perfectly valid options within your constraints.

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