To solve this system, you could do the following steps:
3x+4y=48
6x+2y=60
Choose the best method to solve by looking at the way the equations are set up for you
The best method to solve this system of equations is by using the method of substitution.
1. Start by solving one of the equations for one variable in terms of the other variable. Let's solve the first equation for x:
3x + 4y = 48
3x = 48 - 4y
x = (48 - 4y) / 3
2. Substitute the expression for x in terms of y into the other equation. Let's substitute in the second equation:
6x + 2y = 60
6((48 - 4y) / 3) + 2y = 60
3. Simplify and solve for y:
16 - 8y + 2y = 60
-6y + 16 = 60
-6y = 60 - 16
-6y = 44
y = 44 / -6
y = -22/3 or approximately -7.333
4. Substitute the value of y back into the expression for x to find the corresponding value:
x = (48 - 4(-22/3)) / 3
x = (48 + 88/3) / 3
x = (144/3 + 88/3) / 3
x = (232/3) / 3
x = 232/9 or approximately 25.778
5. So the solution to the system of equations is x = 232/9 and y = -22/3.
To choose the best method to solve this system, we can analyze the equations and consider the structure of the coefficients. In this case, we have:
3x + 4y = 48 (Equation 1)
6x + 2y = 60 (Equation 2)
Since the coefficients of both x and y in Equation 2 are twice the coefficients in Equation 1, it suggests that eliminating one variable through multiplication might be a useful approach.
Based on this observation, the best method to solve this system is the elimination method (also known as the addition/subtraction method). Here's how you can apply this method:
Step 1: Multiply one or both equations by appropriate factors to make the coefficients of either x or y in both equations equal or opposite.
In this case, we can choose to multiply Equation 1 by 2 to make the coefficients of y equal:
2(3x + 4y) = 2(48)
6x + 8y = 96 (Equation 3)
Now, the coefficients of y in both equations are equal.
Step 2: Subtract Equation 3 from Equation 2 to eliminate y.
(6x + 2y) - (6x + 8y) = 60 - 96
6x + 2y - 6x - 8y = -36
-6y = -36
y = (-36) / (-6)
y = 6
Step 3: Substitute the value of y back into either Equation 1 or Equation 2 to solve for x.
Let's substitute y = 6 into Equation 1:
3x + 4(6) = 48
3x + 24 = 48
3x = 48 - 24
3x = 24
x = 24 / 3
x = 8
Step 4: Check the solution by substituting the values of x and y into both original equations.
Substituting x = 8 and y = 6 into Equation 1:
3(8) + 4(6) = 48
24 + 24 = 48
48 = 48 (True)
Substituting x = 8 and y = 6 into Equation 2:
6(8) + 2(6) = 60
48 + 12 = 60
60 = 60 (True)
Both equations are satisfied when x = 8 and y = 6, so the solution is x = 8 and y = 6.