To solve the systems of equations, we can use the method of substitution or elimination.
1) Substitution:
From the first equation, y = 4 - 3x
Substitute this value of y into the second equation: 2x - (4 - 3x) = 6
Simplify: 2x - 4 + 3x = 6
Combine like terms: 5x - 4 = 6
Add 4 to both sides: 5x = 10
Divide by 5: x = 2
Substitute this value of x into the first equation to find y: 3(2) + y = 4
Simplify: 6 + y = 4
Subtract 6 from both sides: y = -2
The solution is (x, y) = (2, -2)
The product is (-2)(2) = -4
2) Elimination:
Multiply the first equation by 2 and the second equation by -1:
4x + 2y = -6
-3x - 2y = 8
Add the two equations: x = 2
Substitute this value of x into the second equation to find y: 3(2) + 2y = -8
Simplify: 6 + 2y = -8
Subtract 6 from both sides: 2y = -14
Divide by 2: y = -7
The solution is (x, y) = (2, -7)
The product is (-7)(2) = -14
3) Elimination:
Multiply the first equation by 5 and the second equation by -4:
35x - 20y = -35
-20x - 4y = 88
Add the two equations: 15x - 24y = 53
Multiply the second equation by 15 and the first equation by -4:
-60x - 12y = -132
-20x + 15y = 75
Add the two equations: -80x + 3y = -57
Multiply the second equation by -16:
320x - 60y = -528
80x - 60y = 300
Subtract the two equations: -400x = -828
Divide by -400: x = 2.07
Substitute this value of x into the first equation to find y: 7(2.07) - 4y = -7
Simplify: 14.49 - 4y = -7
Subtract 14.49 from both sides: -4y = -21.49
Divide by -4: y = 5.37
The solution is (x, y) = (2.07, 5.37)
The product is (5.37)(2.07) = 11.1159
4) Elimination:
Multiply the first equation by -1 and the second equation by 1:
x - 9y = 5
x - 5y = 1
Subtract the second equation from the first equation: -4y = 4
Divide by -4: y = -1
Substitute this value of y into the first equation to find x: x - 9(-1) = 5
Simplify: x + 9 = 5
Subtract 9 from both sides: x = -4
The solution is (x, y) = (-4, -1)
The product is (-1)(-4) = 4
5) Elimination:
Multiply the first equation by -2 and the second equation by -3:
-2x + 2y = 8
6x - 9y = -18
Add the two equations: 4x - 7y = -10
Multiply the second equation by 2:
4x - 6y = 12
4x - 7y = -10
Subtract the two equations: y = 22
Substitute this value of y into the first equation to find x: x - 22 = -4
Add 22 to both sides: x = 18
The solution is (x, y) = (18, 22)
The product is (18)(22) = 396
To find the mean of all five products:
Mean = (-4 + -14 + 11.1159 + 4 + 396)/5 = 78.42318
Solve the following systems of equations. Find the product of each solution (ordered pair -x and y value) and then find the mean of all five products
3x+y=4 and 2x-y=6
2x+y=-3 and 3x+2y=-8
7x-4y=-7 and 5x+y=22
-x+9y=-5 and x-5y=1
x-y=-4 and -2x +3y=6
1 answer
1 answer