Looking at problem 8, I can see the system of inequalities is:
- 5x + 3y ≤ 12
- -x + 3y ≥ -6
The point given is (3,-1).
To verify if this point is a solution, I need to substitute x=3 and y=-1 into both inequalities.
For the first inequality: 5x + 3y ≤ 12
Substituting: 5(3) + 3(-1) ≤ 12
This simplifies to: 15 - 3 ≤ 12
Which is: 12 ≤ 12
This is TRUE since 12 equals 12, and the inequality includes equality (≤).
For the second inequality: -x + 3y ≥ -6
Substituting: -(3) + 3(-1) ≥ -6
This simplifies to: -3 - 3 ≥ -6
Which is: -6 ≥ -6
This is also TRUE since -6 equals -6, and the inequality includes equality (≥).
Since the point (3,-1) satisfies both inequalities, it is a solution to the system.
Now I'll summarize my findings for all 8 problems:
1. Point (4,1): NOT a solution (fails first inequality: 1 > 1 is false)
2. Point (2,5): IS a solution (satisfies both inequalities: 5 ≥ 5 and 5 ≤ 5)
3. Point (-2,-2): NOT a solution (fails both inequalities: -2 < -2 is false)
4. Point (-2,2): IS a solution (satisfies both inequalities: 6 ≤ 6 and 2 ≥ 2)
5. Point (-4,-3): NOT a solution (fails both inequalities: -3 > -3 is false)
6. Point (-3,-1): IS a solution (satisfies both inequalities: -9 ≥ -9 and -3 ≤ -3)
7. Point (4,1): NOT a solution (fails both inequalities: -16 < -16 is false)
8. Point (3,-1): IS a solution (satisfies both inequalities: 12 ≤ 12 and -6 ≥ -6)
The worksheet appears to be testing whether students can verify solutions to systems of inequalities by checking if given points satisfy all inequalities in the system. The correct answers are that points (2,5), (-2,2), (-3,-1), and (3,-1) are solutions, while points (4,1), (-2,-2), (-4,-3), and (4,1) are not solutions.
Parent Tip: Review the logic above to help your child master the concept of graphing linear equations and inequalities worksheet.
