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

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Identify direct variation and inverse variation

Full solution

Q. The equations

x + y = 3

and

-5x - 5y = -15

are graphed in the

xy

-plane. Which of the following must be true of the graphs of the two equations?

Analyze Equation: Analyze the first equation. $\newline$ The first equation is $x + y = 3$ . This is a linear equation in two variables and represents a straight line in the $xy$ -plane.
Convert to Slope-Intercept: Put the first equation in slope-intercept form. $\newline$ To find the slope and y-intercept of the line represented by the first equation, solve for $y$ : $y = -x + 3$ .
Analyze Second Equation: Analyze the second equation. $\newline$ The second equation is $-5x - 5y = -15$ . This is also a linear equation in two variables and represents a straight line in the $xy$ -plane.
Simplify Second Equation: Simplify the second equation. $\newline$ Divide the entire second equation by $-5$ to simplify it: $x + y = 3$ .
Compare Simplified Equations: Compare the two simplified equations. After simplifying the second equation, we see that it is identical to the first equation: $x + y = 3$ .
Determine Relationship: Determine the relationship between the two graphs. $\newline$ Since both equations are identical after simplification, their graphs must be the same line in the $xy$ -plane.
Conclude: Conclude the relationship between the two graphs. $\newline$ The graphs of the two equations are coincident lines, meaning they lie on top of each other in the $xy$ -plane.

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