Use the elimination method to solve each system. How do you do this?
{u+4y=11
7u+3v=2
To solve a system of equations using the elimination method, follow these steps:
Step 1: Write down the given system of equations.
{u + 4y = 11 (Equation 1)
7u + 3v = 2 (Equation 2)
Step 2: Choose one of the variables to eliminate by multiplying one or both equations by a constant so that the coefficients of the variable are the same or opposite in both equations. In this example, we can choose to eliminate "u."
In Equation 1, the coefficient of "u" is 1, and in Equation 2, the coefficient of "u" is 7. To make these coefficients the same, we can multiply Equation 1 by 7 and Equation 2 by 1, or vice versa.
Multiplying Equation 1 by 7: 7(u + 4y) = 7(11) -> 7u + 28y = 77 (Equation 3)
Multiplying Equation 2 by 1: 1(7u + 3v) = 1(2) -> 7u + 3v = 2 (Equation 4)
Step 3: Subtract one equation from the other to eliminate the chosen variable. In this case, we will subtract Equation 4 from Equation 3.
(Equation 3) - (Equation 4): (7u + 28y) - (7u + 3v) = 77 - 2
This simplifies to:
7u - 7u + 28y - 3v = 75
Simplifying further:
25y - 3v = 75 (Equation 5)
Now, we have one equation with "y" and "v" only.
Step 4: Solve the equation obtained in Step 3 for one variable. In this case, we will solve Equation 5 for "v".
25y - 3v = 75
To isolate "v", we will move all terms not containing "v" to the other side of the equation:
-3v = 75 - 25y
Dividing both sides by -3:
v = (75 - 25y) / -3
= (-75 + 25y) / 3 (Equation 6)
Step 5: Substitute the value of "v" from Step 4 into one of the original equations to solve for the remaining variables. In this case, we will substitute the value of "v" from Equation 6 into Equation 2.
7u + 3v = 2
Replacing "v" with (-75 + 25y) / 3:
7u + 3[(-75 + 25y) / 3] = 2
Multiplying both sides of the equation by 3 to eliminate the fraction:
7u - 75 + 25y = 6
Simplifying:
7u + 25y = 81 (Equation 7)
Now, we have one equation with "u" and "y" only.
Step 6: Solve the equation obtained in Step 5 for one variable. In this case, we will solve Equation 7 for "u".
7u + 25y = 81
To isolate "u", we will move all terms not containing "u" to the other side of the equation:
7u = 81 - 25y
Dividing both sides by 7:
u = (81 - 25y) / 7 (Equation 8)
Step 7: Substitute the values of "u" and "v" from Steps 4 and 6, respectively, into one of the original equations to solve for the remaining variable. In this case, we will substitute the values of "u" and "v" from Equations 8 and 6, respectively, into Equation 1.
u + 4y = 11
Replacing "u" with (81 - 25y) / 7 and "v" with (-75 + 25y) / 3:
[(81 - 25y) / 7] + 4y = 11
Multiplying both sides of the equation by 7 to eliminate the fraction:
81 - 25y + 28y = 77
Simplifying:
3y = -4
Dividing both sides of the equation by 3:
y = -4/3
Step 8: Substitute the value of "y" from Step 7 into one of the original equations to solve for the remaining variables. In this case, we will substitute the value of "y" into Equation 1.
u + 4(-4/3) = 11
Simplifying:
u - 16/3 = 11
Adding 16/3 to both sides of the equation:
u = 11 + 16/3
= 49/3
Step 9: Finally, we have the values of all variables. The solution to the system of equations is:
u = 49/3
v = (-75 + 25y) / 3 (from Equation 6)
y = -4/3
So, the solution is (u, v, y) = (49/3, (-75 + 25y) / 3, -4/3).
These are the steps to solve a system of equations using the elimination method. You can apply these steps to any given system to find the solution.