At the research station, Emek and Luna prepared to contact a communications satellite passing overhead. The tracking system displayed two conditions for its position on a coordinate grid: x + y = 13 and 2x - y = 8. The antenna had to point toward coordinates that satisfied both equations. Emek decided to add the equations because the y terms would cancel. This gave 3x = 21, so x = 7. Luna substituted 7 into the first equation and found 7 + y = 13, giving y = 6. Their predicted satellite position was (7, 6). Before aiming the antenna, they checked the coordinates in both conditions. The first gave 7 + 6 = 13, and the second gave 2(7) - 6 = 8. Both statements were true. Confident that they had found the shared solution, Emek and Luna directed the antenna toward (7, 6). Moments later, the station received a clear signal from the satellite.

Challenges

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Challenge 1 of 5

Challenges

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Challenge 1 of 4

Hint

Look at the coefficients of y in the two equations.

Why was adding the equations x + y = 13 and 2x - y = 8 an efficient strategy for finding the satellite's position?

The x terms cancel, leaving an equation with only y.Both x and y cancel, so no substitution is needed.The y terms cancel, leaving the equation 3x = 21.The constants cancel, leaving the equation 3x = 0.

Correct. The terms y and -y cancel when the equations are added.

Not quite. Check which variable has opposite coefficients in the two equations.

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Hint

Add the left sides and the right sides of the two equations. The y terms will cancel.

For another satellite pass, the first tracking condition remains x + y = 13, but the second condition changes to 2x - y = 11. If Emek adds the equations to eliminate y, what integer value does he find for x?

Correct. Adding the equations gives 3x = 24, so x = 8.

Check the result of adding the two equations, then divide to isolate x.

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Hint

Substitute each coordinate pair into both equations. A solution must make both equations true.

Suppose the tracking display lists four possible satellite positions. Which position satisfies both original conditions x + y = 13 and 2x - y = 8?

(6, 7)(7, 6)(8, 5)(9, 4)

Correct. The point (7, 6) satisfies both tracking conditions.

Not quite. A coordinate pair must satisfy both equations, not just one of them.

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Uydu Yörüngesini Bulmak