Question

problems 1–5: here is a coordinate plane.

1. determine which line represents this condition: the coordinates of each point have a sum of 2.
2. determine which line represents this condition: the ( y )-coordinate of each point is 10 less than the ( x )-coordinate.
3. select all the points whose coordinates have a sum of 2.

a. point ( a )
b. point ( b )
c. point ( c )
d. point ( d )
e. point ( e )

1. select all the points whose ( y )-coordinate is 10 less than the ( x )-coordinate.

a. point ( a )
b. point ( b )
c. point ( c )
d. point ( d )
e. point ( e )

1. select all the points whose coordinates have a sum of 2 and the ( y )-coordinate is 10 less than the ( x )-coordinate.

a. point ( a )
b. point ( b )
c. point ( c )
d. point ( d )
e. point ( e )

Explanation:

Step1: Identify point coordinates

From the graph:

• Point $A$: $(1, 4)$
• Point $B$: $(6, -4)$
• Point $C$: $(12, -10)$
• Point $D$: $(2, -8)$
• Point $E$: $(13, 3)$

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Problem 1 Step1: Rewrite condition to line form

Condition: $x + y = 2$, rearrange to $y = -x + 2$. This is a line with slope $-1$ and y-intercept $2$, matching line $n$.

Problem 1 Step2: Match to graph line

Line $n$ follows $x+y=2$.

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Problem 2 Step1: Translate condition to line form

Condition: $y = x - 10$. This is a line with slope $1$ and y-intercept $-10$, matching line $m$.

Problem 2 Step2: Match to graph line

Line $m$ follows $y=x-10$.

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Problem 3 Step1: Test sum $x+y=2$ for each point

• $A$: $1+4=5

eq2$

• $B$: $6+(-4)=2$
• $C$: $12+(-10)=2$
• $D$: $2+(-8)=-6

eq2$

• $E$: $13+3=16

eq2$

Problem 3 Step2: Select valid points

Points $B, C$ satisfy the condition.

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Problem 4 Step1: Test $y=x-10$ for each point

• $A$: $4=1-10\Rightarrow4=-9$ (False)
• $B$: $-4=6-10\Rightarrow-4=-4$ (True)
• $C$: $-10=12-10\Rightarrow-10=2$ (False)
• $D$: $-8=2-10\Rightarrow-8=-8$ (True)
• $E$: $3=13-10\Rightarrow3=3$ (True)

Problem 4 Step2: Select valid points

Points $B, D, E$ satisfy the condition.

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Problem 5 Step1: Find points satisfying both conditions

Need $x+y=2$ and $y=x-10$. Substitute $y=x-10$ into $x+y=2$:
$x+(x-10)=2\Rightarrow2x=12\Rightarrow x=6$, then $y=6-10=-4$. This is Point $B$.

Problem 5 Step2: Verify against points

Only Point $B$ meets both rules.

Answer:

1. Line $n$
2. Line $m$
3. B. Point B, C. Point C
4. B. Point B, D. Point D, E. Point E
5. B. Point B