Solve by elimination method.
2r-5s=-33
5r+2s=48
Solve by substitution method.
5x+6y=5
x-9y=40
I always get these two confused... can someone please help me with these two problems?? I would really appreciate it..
Sure! I can help you solve these two systems of equations using both the elimination and substitution methods. Let's start with the elimination method for the first problem:
1. Elimination method:
Step 1: Multiply the first equation by 5 and the second equation by 2 to make the coefficients of 's' in both equations opposite in sign.
Equation 1: 10r - 25s = -165
Equation 2: 10r + 4s = 96
Step 2: Subtract the second equation from the first equation to eliminate 'r' and solve for 's'.
(10r - 25s) - (10r + 4s) = -165 - 96
-29s = -261
Divide both sides by -29 to get the value of 's':
s = -261 / -29
Thus, s = 9.
Step 3: Substitute the value of 's' into either of the original equations to solve for 'r'.
Using first equation:
2r - 5(9) = -33
2r - 45 = -33
2r = -33 + 45
2r = 12
Divide both sides by 2 to get the value of 'r':
r = 12 / 2
Thus, r = 6.
Therefore, the solution to the system of equations is r = 6 and s = 9.
Now let's move on to the substitution method for the second problem:
2. Substitution method:
Step 1: Solve one equation for one variable in terms of the other variable.
From the second equation, solve for x:
x = 40 + 9y.
Step 2: Substitute the expression for 'x' in terms of 'y' into the first equation.
5(40 + 9y) + 6y = 5
Distribute:
200 + 45y + 6y = 5
Combine like terms:
51y + 200 = 5
Subtract 200 from both sides:
51y = -195
Divide both sides by 51:
y = -195 / 51
Thus, y = -3.8235 (rounded to 4 decimal places).
Step 3: Substitute the value of 'y' back into the equation for 'x':
x = 40 + 9(-3.8235)
x = 40 - 34.4115
x = 5.5885 (rounded to 4 decimal places).
Therefore, the solution to the system of equations is x = 5.5885 and y = -3.8235.