6x-3y = -6 and -5x + 7y = 41
To solve this system of equations, we'll use the method of substitution. Here's how to do it step by step:
Step 1: Solve one equation for one variable.
Let's start by solving the first equation, 6x - 3y = -6, for x. You can do this by isolating x on one side of the equation:
6x - 3y = -6
Add 3y to both sides:
6x = 3y - 6
Divide both sides by 6:
x = (3y - 6)/6
Step 2: Substitute the expression for x into the other equation.
Now, substitute the expression we found for x into the second equation, -5x + 7y = 41:
-5((3y - 6)/6) + 7y = 41
Step 3: Solve for y.
Simplify the equation by multiplying both sides by 6 to eliminate the denominator:
-5(3y - 6) + 42y = 246
Distribute -5 on the left side:
-15y + 30 + 42y = 246
Combine like terms:
27y + 30 = 246
Subtract 30 from both sides:
27y = 216
Divide both sides by 27:
y = 8
Step 4: Solve for x.
Now that we have y = 8, we can substitute this value back into one of the original equations, let's use the first equation, 6x - 3y = -6:
6x - 3(8) = -6
6x - 24 = -6
Add 24 to both sides:
6x = 18
Divide both sides by 6:
x = 3
Step 5: Check the solution.
Finally, check the solution by substituting the values of x and y into the second equation, -5x + 7y = 41:
-5(3) + 7(8) = 41
-15 + 56 = 41
41 = 41
The solution to the system of equations is x = 3 and y = 8.
To solve the system of equations:
6x - 3y = -6 ......(1)
-5x + 7y = 41 ......(2)
We can solve this system of equations using the method of elimination or substitution. I will use the method of substitution.
Step 1: Solve one equation for one variable in terms of the other variable.
From equation (1) we have:
6x - 3y = -6
Rearranging, we get:
6x = 3y - 6
Dividing by 6, we get:
x = (3y - 6)/6
Simplifying, we get:
x = (y - 2)/2 ......(3)
Step 2: Substitute the expression for x from step 1 into the second equation.
Substituting (y - 2)/2 for x in equation (2), we get:
-5((y - 2)/2) + 7y = 41
Multiply through by 2 to remove the fraction:
-5(y - 2) + 14y = 82
Expand and simplify:
-5y + 10 + 14y = 82
Combine like terms:
9y + 10 = 82
Subtract 10 from both sides:
9y = 72
Divide by 9:
y = 8
Step 3: Substitute the value of y into equation (3) to find x.
Using equation (3):
x = (y - 2)/2
Substituting y = 8, we have:
x = (8 - 2)/2
Simplifying, we get:
x = 6/2
x = 3
The solution to the system of equations is x = 3 and y = 8.