Solve by using Elimination method.
5r-6s=29
6r+5s=47
multiply the first equation by 5, the second equation by 6, then ADD the equations. You will eliminate the coefficents of s.
To solve the given system of equations using the Elimination method, follow these steps:
1. Multiply one or both equations by a constant, if necessary, to make the coefficients of one of the variables in both equations equal or opposite. In this case, by multiplying the first equation by 6 and the second equation by 5, the coefficients of 's' in both equations will become equal and opposite.
Multiplying the first equation by 6:
(6)(5r - 6s) = (6)(29) becomes 30r - 36s = 174
Multiplying the second equation by 5:
(5)(6r + 5s) = (5)(47) becomes 30r + 25s = 235
2. Now, add the two equations together to eliminate one of the variables. In this case, when we add the equations, the 'r' terms will cancel out and we will be left with an equation only involving 's'.
(30r - 36s) + (30r + 25s) = 174 + 235 becomes 60r - 11s = 409
3. Solve this equation for 's' by isolating the variable. Subtract 60r from both sides:
60r - 11s = 409
-11s = 409 - 60r
-11s = -60r + 409
4. Divide both sides of the equation by -11 to solve for 's':
s = (-60r + 409) / -11
s = (-60/-11)r + (409/-11)
s = 60/11r - 37
5. Substitute the value of 's' in one of the original equations to solve for 'r'. Let's use the first equation:
5r - 6s = 29
Substitute s with 60/11r - 37:
5r - 6(60/11r - 37) = 29
5r - (360/11r - 222) = 29
5r - 360/11r + 222 = 29
6. Combine like terms and simplify the equation:
(55r^2 - 3200r + 2007) / 11r = 29
55r^2 - 3200r + 2007 = 29 * 11r
55r^2 - 3200r + 2007 = 319r
55r^2 - 3519r + 2007 = 0
7. Solve the quadratic equation by factoring, completing the square, or using the quadratic formula. Since this equation does not easily factor, we'll use the quadratic formula:
r = (-b ± √(b^2 - 4ac)) / (2a)
a = 55, b = -3519, c = 2007
r = (-(-3519) ± √((-3519)^2 - 4(55)(2007))) / (2(55))
r = (3519 ± √(12378561 - 440220)) / 110
r = (3519 ± √11938341) / 110
8. Use a calculator to find the approximate values of 'r':
r ≈ (3519 + √11938341) / 110 or r ≈ (3519 - √11938341) / 110
After obtaining the values of 'r', substitute them back into any of the original equations to solve for 's'.