13x+22y=16
32x+12y=22
13x+22y=16....(i)
32x+12y=22....(ii)
eqn (ii) can be simplified by dividing through by 2, which gives 16x+6y=11
so we have;
13x+22y=16
16x+6y=11
by using elimination method, multiply eqn (i) by 3 and eqn (ii) by 11. So we have;
39x+66y=48
176x+66y=121
subtract eqn (i) from eqn (ii), ie,
(176x+66y=121)- (39x+66y=48)
so, we have;
137x=73 (divide both sides by 137)
x=73/137
substitute x=73/137 to the eqn 16x+6y=11
(16×73/137)+ 6y=11
1168/137+ 6y=11
6y=11-1168/137
6y=339/137 (divide both sides by 6)
y=339/822
therefore,
x=73/137 and y=339/822
To find the values of 'x' and 'y' that satisfy the system of equations:
1. Start by writing the system of equations:
13x + 22y = 16 -- (Equation 1)
32x + 12y = 22 -- (Equation 2)
2. There are various methods to solve a system of equations, but let's use the method of substitution.
3. Solve one of the equations for one variable in terms of the other. Let's solve Equation 1 for 'x':
13x = 16 - 22y
Divide both sides of the equation by 13:
x = (16 - 22y) / 13 -- (Equation 3)
4. Substitute the expression for 'x' from Equation 3 into Equation 2:
32((16 - 22y) / 13) + 12y = 22
5. Simplify the equation and solve for 'y':
Multiply both sides of the equation by 13 to eliminate the denominator:
32(16 - 22y) + 156y = 286
Distribute the 32 on the left side:
512 - 704y + 156y = 286
Combine like terms:
-548y + 512 = 286
Move constants to the right side:
-548y = 286 - 512
-548y = -226
Divide both sides of the equation by -548:
y = (-226) / (-548)
Simplify:
y ≈ 0.4124
6. Substitute the value of 'y' back into Equation 3 to find 'x':
x = (16 - 22(0.4124)) / 13
Simplify:
x ≈ 0.804
7. The solution to the system of equations is approximately:
x ≈ 0.804
y ≈ 0.4124
To solve the system of equations, we can use the method of substitution or the method of elimination. Let's use the method of elimination.
Step 1: Multiply the first equation by 32 and the second equation by 13 to make the coefficients of x in both equations equal.
32 * (13x + 22y) = 32 * 16
13 * (32x + 12y) = 13 * 22
This simplifies to:
416x + 704y = 512
416x + 156y = 286
Step 2: Now subtract the second equation from the first equation to eliminate x.
(416x + 704y) - (416x + 156y) = 512 - 286
Simplifying further:
548y = 226
Step 3: Solve for y by dividing both sides of the equation by 548.
y = 226 / 548
Simplifying the fraction:
y = 113 / 274
Step 4: Now substitute the value of y back into any of the original equations. Let's use the first equation.
13x + 22 * (113 / 274) = 16
Simplifying further:
13x + 2476 / 274 = 16
To simplify the equation further, we can multiply every term by 274 to get rid of the fraction:
274 * 13x + 2476 = 16 * 274
Simplifying more:
3562x + 2476 = 4384
Step 5: Solve for x by isolating the variable.
3562x = 4384 - 2476
3562x = 1908
Divide both sides of the equation by 3562 to solve for x:
x = 1908 / 3562
Step 6: Simplify the fraction:
x = 954 / 1781
So, the solution to the system of equations is:
x = 954 / 1781
y = 113 / 274