Solving the simultaneous equations shows the following solutions:
a) x = 3/8 and y = -25/16
b) x = 9 and y = 3
c) x = 6 and y ≈ 0.3636…
d) x = 7 and y = 4.
Simultaneous equations are systems of equations solved concurrently.
There are various methods of solving simultaneous equations. We can solve simultaneous equations using the elimination method, substitution method, graphing method, and matrix method.
a) 3x = -2y -7
5x -2y + 11 = 10
Solving the first equation:
y = (-3/2)x - (7/2)
Substituting this expression for y into the second equation:
5x - 2((-3/2)x - (7/2)) + 11 = 10
Simplifying this expression:
5x + 3x + 7 = 10
Combining like terms:
8x = 3
Dividing both sides by 8:
x = 3/8
Substituting this value of x into the first equation:
3(3/8) = -2y - 7
Simplifying this expression:
y = -25/16
Thus, the solution to the system of equations is x = 3/8 and y = -25/16.
b) 69 = -2y + 7x
4x = 3y + 45
Solving the second equation for y:
y = (4/3)x - 15
Substituting this expression for y into the first equation:
69 = -2((4/3)x - 15) + 7x
Simplifying this expression:
69 = (-8/3)x + 30 + 7x
Combining like terms:
69 = (13/3)x + 30
Subtracting 30 from both sides:
39 = (13/3)x
Multiplying both sides by 3/13:
x = 9
Substituting this value of x into the second equation:
4(9) = 3y + 45
Simplifying this expression:
y = 3
Thus, the solution to the system of equations is x = 9 and y = 3.
c) 0.55x - 0.22y = 2.2
2.5x - 0.6y = 2
Solving the first equation for y:
y = (5/11)x - (20/11)
Substituting this expression for y into the second equation:
2.5x - 0.6((5/11)x - (20/11)) = 2
Simplifying this expression:
2.5x + 3x - 12 = 22
Combining like terms:
5.5x = 34
Dividing both sides by 5.5:
x = 6
Substituting this value of x into the first equation:
0.55(6) - 0.22y = 2.2
Simplifying this expression:
-0.22y = -0.08
Dividing both sides by -0.22:
y = 0.3636…
Thus, the solution to the system of equations is x = 6 and y ≈ 0.3636…
d) 1/2x + 1/4y = 9/2
13 + 2y = 3x
Solving the first equation for y:
y = 18 - 2x
Substituting this expression for y into the second equation:
13 + 2(18 - 2x) = 3x
Simplifying this expression:
49 = 7x
Dividing both sides by 7:
x = 7
Substituting this value of x into the first equation:
1/2(7) + 1/4y = 9/2
Simplifying this expression:
7/2 + 1/4y = 9/2
Subtracting 7/2 from both sides:
1/4y = 1
Multiplying both sides by 4:
y = 4
Thus, the solution to the system of equations is x = 7 and y = 4.
Complete Question:
Solve the following simultaneous equations:
a) 3x = -2y -7
5x -2y + 11 = 10
b) 69 = -2y + 7x
4x = 3y + 45
c) 0.55x - 0.22y = 2.2
2.5x - 0.6y = 2
d) 1/2x + 1/4y = 9/2
13 + 2y = 3x