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The equations x+y=3x + y = 3 and −5x−5y=−15-5x - 5y = -15 are graphed in the xyxy-plane. Which of the following must be true of the graphs of the two equations?

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Identify direct variation and inverse variationright chevron icon

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Q. The equations x+y=3x + y = 3 and −5x−5y=−15-5x - 5y = -15 are graphed in the xyxy-plane. Which of the following must be true of the graphs of the two equations?
  1. Analyze Equation: Analyze the first equation.\newlineThe first equation is x+y=3x + y = 3. This is a linear equation in two variables and represents a straight line in the xyxy-plane.
  2. Convert to Slope-Intercept: Put the first equation in slope-intercept form.\newlineTo find the slope and y-intercept of the line represented by the first equation, solve for yy: y=−x+3y = -x + 3.
  3. Analyze Second Equation: Analyze the second equation.\newlineThe second equation is −5x−5y=−15-5x - 5y = -15. This is also a linear equation in two variables and represents a straight line in the xyxy-plane.
  4. Simplify Second Equation: Simplify the second equation.\newlineDivide the entire second equation by −5-5 to simplify it: x+y=3x + y = 3.
  5. Compare Simplified Equations: Compare the two simplified equations. After simplifying the second equation, we see that it is identical to the first equation: x+y=3x + y = 3.
  6. Determine Relationship: Determine the relationship between the two graphs.\newlineSince both equations are identical after simplification, their graphs must be the same line in the xyxy-plane.
  7. Conclude: Conclude the relationship between the two graphs.\newlineThe graphs of the two equations are coincident lines, meaning they lie on top of each other in the xyxy-plane.

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