Explain why the addition method might be preferred over the substitution method for solving the system
2x-3y=5
5x+2y=6
Two simple multiplications, and you can do it in your head.
multiply top equation by 2
4x-6y=10
multiply the bottom equation by 3
15x+6y=18
add 19x=38
x=2
Thank you this does help. But why is the addition method preferred over the substitution method for solving the system
2x-3y=5
5x+2y=6
I got how you got the 19x=38 but whatis the x=2(sorry not good with algebra)
The addition method, also known as the elimination method, might be preferred over the substitution method for solving this system of equations because it involves eliminating one of the variables by adding or subtracting the equations together. In this particular case, the addition method is advantageous because the coefficients of y in both equations have opposite signs (-3y and 2y).
To solve the system using the addition method, we need to manipulate the equations so that when we add them together, one of the variables is eliminated. Let's start with the original equations:
Equation 1: 2x - 3y = 5
Equation 2: 5x + 2y = 6
To eliminate the y variable, we can multiply Equation 1 by 2 and Equation 2 by 3:
Equation 1 (multiplied by 2): 4x - 6y = 10
Equation 2 (multiplied by 3): 15x + 6y = 18
Now, we can add the two equations together:
(4x - 6y) + (15x + 6y) = 10 + 18
Simplifying, we have:
19x = 28
Now, we can isolate x by dividing both sides of the equation by 19:
x = 28/19
To find the value of y, we substitute this value of x back into either of the original equations. Let's use Equation 1:
2(28/19) - 3y = 5
Multiplying, we get:
56/19 - 3y = 5
Now, subtracting 56/19 from both sides:
-3y = 5 - 56/19
Simplifying, we have:
-3y = (95 - 56)/19
-3y = 39/19
Dividing by -3:
y = (39/19) * (-1/3)
Simplifying further, we have:
y = -13/19
Therefore, the solution to the system of equations is x = 28/19 and y = -13/19. The addition method was used in this case because it facilitated the elimination of the y variable by adding the equations together.
The addition method, also known as the elimination method or the method of adding equations, is often preferred over the substitution method for solving a system of equations when both equations are in standard form (ax + by = c) and involve coefficients that are relatively easy to manipulate. This method is especially useful when the coefficients of one variable in the two equations are additive inverses, meaning that their sum is equal to zero.
In the given system:
2x - 3y = 5
5x + 2y = 6
To use the addition method, you start by manipulating the coefficients of one variable so that they become additive inverses in the two equations. In this case, if we multiply the first equation by 5 and the second equation by 2, we can make the coefficients of "x" in both equations have a sum of zero:
10x - 15y = 25
10x + 4y = 12
Now, we can add the equations together vertically, eliminating the "x" variable. The resulting equation will only have the "y" variable:
-15y + 4y = 25 + 12
-11y = 37
From here, we can solve for "y" by dividing both sides of the equation by -11:
y = -37/11 = -3.3636 (rounded to four decimal places)
Now that we have found the value of "y," we can substitute it back into one of the original equations to solve for "x." Let's use the first equation:
2x - 3(-3.3636) = 5
2x + 10.0908 = 5
2x = -5.0908
x = -5.0908/2 = -2.5454 (rounded to four decimal places)
Hence, the solution to the system of equations is x = -2.5454 and y = -3.3636.
The addition method is often preferred over the substitution method in this case because it allows us to eliminate one variable (in this case, "x") easily by adding the equations together, removing the need to solve for one variable and then substitute that value back into the other equation. This method can often save time and effort in solving a system of equations.