Solve the system for
2x=3y=2
3x-5y=22
well, assuming your = means +,
6x+9y=6
6x-10y=44
...
To solve the system of equations:
1) Simplify the first equation:
2x = 3y + 2
2) Rearrange the equation to isolate one variable (x or y). Let's isolate x in this case:
2x - 3y = 2
3) Simplify the second equation:
3x - 5y = 22
4) Now, we have a system of two equations:
2x - 3y = 2
3x - 5y = 22
We can solve this system using the substitution method or the elimination method. Let's use the elimination method:
5) Multiply the first equation by 3 and the second equation by 2 to create opposite coefficients for y:
(3)(2x - 3y) = (3)(2)
(2)(3x - 5y) = (2)(22)
6) Simplify:
6x - 9y = 6
6x - 10y = 44
7) Subtract the new equations to eliminate x:
(6x - 9y) - (6x - 10y) = 6 - 44
6x - 9y - 6x + 10y = -38
y = -38
8) Substitute the value of y (-38) into one of the original equations. Let's use the first equation:
2x - 3(-38) = 2
2x + 114 = 2
2x = 2 - 114
2x = -112
9) Divide both sides by 2 to find x:
x = -112/2
x = -56
So, the solution to the system of equations is x = -56 and y = -38.
To solve the system of equations:
Step 1: Write the system of equations:
2x - 3y = 2 ...(Equation 1)
3x - 5y = 22 ...(Equation 2)
Step 2: Choose a method to solve the system. In this case, we can use either the substitution method or the elimination method. Let's use the elimination method.
Step 3: Multiply both sides of Equation 1 by 3 to make the coefficients of y the same:
(3)(2x - 3y) = (3)(2)
6x - 9y = 6 ...(Equation 3)
Step 4: Now, we have two equations with coefficients of y that are equal:
6x - 9y = 6 ...(Equation 3)
3x - 5y = 22 ...(Equation 2)
Step 5: Subtract Equation 3 from Equation 2 to eliminate y:
(3x - 5y) - (6x - 9y) = 22 - 6
3x - 5y - 6x + 9y = 16
-3x + 4y = 16 ...(Equation 4)
Step 6: Now we have two equations with coefficients of x and y:
-3x + 4y = 16 ...(Equation 4)
2x - 3y = 2 ...(Equation 1)
Step 7: Multiply both sides of Equation 4 by 2, and multiply both sides of Equation 1 by 3 to make the coefficients of x the same:
(2)(-3x + 4y) = (2)(16)
-6x + 8y = 32 ...(Equation 5)
(3)(2x - 3y) = (3)(2)
6x - 9y = 6 ...(Equation 6)
Step 8: Add Equation 5 to Equation 6 to eliminate x:
(-6x + 8y) + (6x - 9y) = 32 + 6
-6x + 6x + 8y - 9y = 38
-y = 38
Step 9: Divide both sides of the equation by -1 to solve for y:
y = -38
Step 10: Substitute the value of y back into either Equation 1 or Equation 2. Let's substitute it into Equation 1:
2x - 3(-38) = 2
2x + 114 = 2
Step 11: Simplify and solve for x:
2x = 2 - 114
2x = -112
x = -56
Step 12: The solution to the system is x = -56 and y = -38.