The point (5, −3)
is a solution to which system of equations?
y=x−8
2x+y=7
y = x + 2
y = x + 5
y = x + 2 y = x + 5
y −12x + 6
y = 3x − 1
y = −12x + 6
y = 3x − 1
y = 23x + 6
3y + 6x − 18 = 0
The solution (5, -3) means that when x = 5, y = -3 satisfies the system of equations. To check which system of equations this point is a solution to, we can substitute x = 5 and y = -3 into each equation and see if the equation is true:
1. y = x - 8: -3 = 5 - 8, this equation is true with (5, -3)
2. 2x + y = 7: 2(5) + (-3) = 7, this equation is true with (5, -3)
3. y = x + 2: -3 = 5 + 2, this equation is not true with (5, -3)
4. y = x + 5: -3 = 5 + 5, this equation is not true with (5, -3)
5. y = x + 2 and y = x + 5: -3 = 5 + 2 and -3 = 5 + 5, neither equation is true with (5, -3)
6. y - 12x + 6: -3 - 12(5) + 6 = -69, this equation is not true with (5, -3)
7. y = 3x - 1: -3 = 3(5) - 1, this equation is not true with (5, -3)
8. y = -12x + 6: -3 = -12(5) + 6, this equation is true with (5, -3)
9. y = 23x + 6: -3 = 23(5) + 6, this equation is not true with (5, -3)
10. 3y + 6x - 18 = 0: 3(-3) + 6(5) - 18 = 0, this equation is true with (5, -3)
Therefore, the point (5, -3) is a solution to systems of equations 1, 2, 8, and 10.
To determine which system of equations the point (5, -3) is a solution to, we will substitute the x and y values of the point into each system of equations and see which one results in both equations being true.
1. y = x - 8
Substituting x = 5 and y = -3:
-3 = 5 - 8
-3 = -3 (True)
2. 2x + y = 7
Substituting x = 5 and y = -3:
2(5) + (-3) = 7
10 - 3 = 7
7 = 7 (True)
3. y = x + 2 y = x + 5
Substituting x = 5 and y = -3 into both equations:
-3 = 5 + 2
-3 = 7 (False)
-3 = 5 + 5
-3 = 10 (False)
4. y - 12x + 6 = 0
Substituting x = 5 and y = -3:
-3 - 12(5) + 6 = 0
-3 - 60 + 6 = 0
-57 + 6 = 0
-51 = 0 (False)
5. y = 3x - 1
Substituting x = 5 and y = -3:
-3 = 3(5) - 1
-3 = 15 - 1
-3 = 14 (False)
6. y = -12x + 6
Substituting x = 5 and y = -3:
-3 = -12(5) + 6
-3 = -60 + 6
-3 = -54 (False)
7. y = 3x - 1
Substituting x = 5 and y = -3:
-3 = 23(5) + 6
-3 = 115 + 6
-3 = 121 (False)
8. 3y + 6x - 18 = 0
Substituting x = 5 and y = -3:
3(-3) + 6(5) - 18 = 0
-9 + 30 - 18 = 0
3 + 12 - 18 = 0
15 - 18 = 0
-3 = 0 (False)
Based on the substitutions made, we can see that the point (5, -3) is a solution to the system of equations:
1. y = x - 8
2. 2x + y = 7
To determine which system of equations the point (5, -3) is a solution to, we will substitute the x and y values into each equation and check if they satisfy the equation.
1. y = x - 8
Substituting x = 5 and y = -3:
-3 = 5 - 8
-3 = -3
The point (5, -3) satisfies the equation y = x - 8.
2. 2x + y = 7
Substituting x = 5 and y = -3:
2(5) + (-3) = 7
10 - 3 = 7
7 = 7
The point (5, -3) satisfies the equation 2x + y = 7.
3. y = x + 2
Substituting x = 5 and y = -3:
-3 = 5 + 2
-3 = 7
The point (5, -3) does not satisfy the equation y = x + 2.
4. y = x + 5
Substituting x = 5 and y = -3:
-3 = 5 + 5
-3 = 10
The point (5, -3) does not satisfy the equation y = x + 5.
5. y = x + 2 and y = x + 5
Substituting x = 5 and y = -3:
-3 = 5 + 2 and -3 = 5 + 5
-3 = 7 and -3 = 10
The point (5, -3) satisfies neither equation y = x + 2 nor y = x + 5.
6. y - 12x + 6
Substituting x = 5 and y = -3:
-3 - 12(5) + 6
-3 - 60 + 6
-57 = 0
The point (5, -3) does not satisfy the equation y - 12x + 6.
7. y = 3x - 1
Substituting x = 5 and y = -3:
-3 = 3(5) - 1
-3 = 15 - 1
-3 = 14
The point (5, -3) does not satisfy the equation y = 3x - 1.
8. y = -12x + 6
Substituting x = 5 and y = -3:
-3 = -12(5) + 6
-3 = -60 + 6
-3 = -54
The point (5, -3) does not satisfy the equation y = -12x + 6.
9. y = 3x - 1
Substituting x = 5 and y = -3:
-3 = 3(5) - 1
-3 = 15 - 1
-3 = 14
The point (5, -3) does not satisfy the equation y = 3x - 1.
10. 3y + 6x - 18 = 0
Substituting x = 5 and y = -3:
3(-3) + 6(5) - 18 = 0
-9 + 30 - 18 = 0
21 - 18 = 0
3 = 0
The point (5, -3) does not satisfy the equation 3y + 6x - 18 = 0.
From the above analysis, we can conclude that the point (5, -3) is a solution to the system of equations:
1. y = x - 8
2. 2x + y = 7