Solve the system of equations 3x +6y=21. -5x+2y=13
To solve the system of equations:
3x + 6y = 21 ...(Equation 1)
-5x + 2y = 13 ...(Equation 2)
We can solve this system of equations using the method of substitution or elimination.
Let's use the method of substitution:
From Equation 1, we can express x in terms of y:
3x = 21 - 6y
Dividing both sides by 3:
x = 7 - 2y/3 ...(Equation 3)
Now, substitute x = 7 - 2y/3 into Equation 2:
-5(7 - 2y/3) + 2y = 13
Expanding:
-35 + 10y/3 + 2y = 13
Multiplying both sides by 3 to eliminate the fraction:
-105 + 10y + 6y = 39
Combining like terms:
16y = 144
Dividing both sides by 16:
y = 9
Now, substitute y = 9 back into Equation 3 to find x:
x = 7 - 2(9)/3
x = 7 - 6
x = 1
Therefore, the solution to the system of equations is x = 1 and y = 9.
To solve the system of equations:
1. Start by multiplying the first equation by 2 and the second equation by 3 to eliminate the variable y.
Multiply the first equation: 2(3x + 6y) = 2(21) => 6x + 12y = 42
Multiply the second equation: 3(-5x + 2y) = 3(13) => -15x + 6y = 39
2. Next, we can subtract the second equation from the first equation to eliminate the variable y.
(6x + 12y) - (-15x + 6y) = 42 - 39
Simplify: 6x + 12y + 15x - 6y = 3
Combine like terms: 21x + 6y = 3
3. Now, let's solve for y by isolating it in one of the equations.
Rearrange the equation 3x + 6y = 21 to solve for y:
6y = 21 - 3x
Divide both sides by 6: y = (21 - 3x)/6
Simplify: y = 7/2 - x/2
4. Substitute the value of y in terms of x into the equation 21x + 6y = 3 and solve for x.
Replace y with (7/2 - x/2) in the equation:
21x + 6(7/2 - x/2) = 3
Simplify: 21x + 21 - 3x = 3
Combine like terms: 18x + 21 = 3
Rearrange the equation: 18x = 3 - 21
Simplify: 18x = -18
Divide both sides by 18: x = -1
5. Now, substitute the value of x = -1 into the equation y = 7/2 - x/2 to solve for y.
y = 7/2 - (-1)/2
Simplify: y = 7/2 + 1/2
Combine the fractions: y = 8/2
Simplify: y = 4
Therefore, the solution to the system of equations 3x + 6y = 21 and -5x + 2y = 13 is x = -1 and y = 4.
To solve the system of equations 3x + 6y = 21 and -5x + 2y = 13, we can use the method of substitution. Here's how to go about it:
Step 1: Choose one of the equations and solve it for one variable in terms of the other variable. Let's choose -5x + 2y = 13.
Solving for x, we get:
-5x = 13 - 2y
Dividing both sides by -5, we find:
x = (13 - 2y) / -5
Step 2: Substitute the value of x from the previous step into the other equation in the system. We'll use 3x + 6y = 21.
Substituting x, we have:
3((13 - 2y) / -5) + 6y = 21
Step 3: Simplify and solve for y.
Expanding the equation, we get:
(39 - 6y) / -5 + 6y = 21
To remove the fraction, we multiply every term in the equation by -5:
(39 - 6y) + (-5)(6y) = 21(-5)
39 - 6y - 30y = -105
Combine like terms:
39 - 36y = -105
Subtract 39 from both sides:
-36y = -144
Now, divide both sides by -36:
y = (-144) / (-36)
y = 4
Step 4: Substitute the value of y back into one of the original equations to find x. Let's use 3x + 6y = 21.
Substituting y = 4, we have:
3x + 6(4) = 21
3x + 24 = 21
Subtract 24 from both sides:
3x = -3
Divide both sides by 3:
x = -1
Therefore, the solution to the system of equations is x = -1 and y = 4.