Which set of coordinates satisfies the equations $3x - 2y = 15$ and $4x - y = 20$?
A. $(2, -7)$
B. $(1, -6)$
C. $(5, 0)$
D. $(0, -7.5)$
Answer (2)
The coordinate pair (5, 0) satisfies both equations 3x - 2y = 15 and 4x - y = 20, while the other options do not. Therefore, the correct answer is (5, 0).
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Substitute each coordinate pair into the given equations.
Check if both equations are satisfied for each pair.
The coordinate pair ( 5 , 0 ) satisfies both equations: 3 ( 5 ) − 2 ( 0 ) = 15 and 4 ( 5 ) − ( 0 ) = 20 .
Therefore, the solution is ( 5 , 0 ) .
Explanation
Understanding the Problem We are given two linear equations: 3 x − 2 y = 15 and 4 x − y = 20 . We need to find which of the given coordinate pairs satisfies both equations.
Testing the Coordinates Let's test each coordinate pair to see if it satisfies both equations.
Testing Option A A. ( 2 , − 7 ) :
Substitute x = 2 and y = − 7 into the first equation: 3 ( 2 ) − 2 ( − 7 ) = 6 + 14 = 20 . This does not equal 15, so the first equation is not satisfied. Substitute x = 2 and y = − 7 into the second equation: 4 ( 2 ) − ( − 7 ) = 8 + 7 = 15 . This does not equal 20, so the second equation is not satisfied. Therefore, ( 2 , − 7 ) is not a solution.
Testing Option B B. ( 1 , − 6 ) :
Substitute x = 1 and y = − 6 into the first equation: 3 ( 1 ) − 2 ( − 6 ) = 3 + 12 = 15 . The first equation is satisfied. Substitute x = 1 and y = − 6 into the second equation: 4 ( 1 ) − ( − 6 ) = 4 + 6 = 10 . This does not equal 20, so the second equation is not satisfied. Therefore, ( 1 , − 6 ) is not a solution.
Testing Option C C. ( 5 , 0 ) :
Substitute x = 5 and y = 0 into the first equation: 3 ( 5 ) − 2 ( 0 ) = 15 − 0 = 15 . The first equation is satisfied. Substitute x = 5 and y = 0 into the second equation: 4 ( 5 ) − ( 0 ) = 20 − 0 = 20 . The second equation is satisfied. Therefore, ( 5 , 0 ) is a solution.
Testing Option D D. ( 0 , − 7.5 ) :
Substitute x = 0 and y = − 7.5 into the first equation: 3 ( 0 ) − 2 ( − 7.5 ) = 0 + 15 = 15 . The first equation is satisfied. Substitute x = 0 and y = − 7.5 into the second equation: 4 ( 0 ) − ( − 7.5 ) = 0 + 7.5 = 7.5 . This does not equal 20, so the second equation is not satisfied. Therefore, ( 0 , − 7.5 ) is not a solution.
Final Answer Only the coordinate pair ( 5 , 0 ) satisfies both equations.
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business, calculating the optimal mix of ingredients in a recipe, or modeling traffic flow in a city. In this case, we found the solution to a system of two linear equations, which represents the point where two lines intersect on a graph. This concept is fundamental in fields like economics, engineering, and computer science, where finding the intersection of multiple constraints is crucial for decision-making and problem-solving. For example, businesses use systems of equations to optimize production costs and maximize profits by finding the intersection of cost and revenue functions.
