Explain why the addition method might be preferred over the substitution method for solving the system 2x-3y=5
5x+2y=6
Am lost on these.Please help
To understand why the addition method might be preferred over the substitution method for solving this system, let's first review what each method entails.
1. Substitution Method: In this method, you solve one equation for one variable and substitute that expression into the other equation. By doing this, you eliminate one variable and solve for the remaining variable.
2. Addition Method (also known as the Elimination Method): In this method, you manipulate the equations by adding or subtracting them to eliminate one variable. By doing this, you can solve for the remaining variable.
Now, why might the addition method be preferred over the substitution method for this particular system?
One reason is that the coefficients of the variables in both equations are relatively small and simple in the addition method, making it easier to manipulate the equations and eliminate a variable.
Let's go through the steps using the addition method to solve this system:
1. Arrange the equations so that the variables line up vertically:
2x - 3y = 5
5x + 2y = 6
2. Multiply both equations by appropriate coefficients to get the coefficients of either x or y to be opposites. In this case, we can multiply the first equation by 5 and the second equation by 2:
10x - 15y = 25
10x + 4y = 12
3. Now, subtract the second equation from the first equation:
(10x - 15y) - (10x + 4y) = 25 - 12
10x - 15y - 10x - 4y = 13
-19y = 13
4. Solve for y by dividing both sides of the equation by -19:
-19y / -19 = 13 / -19
y = -13/19
5. Now, substitute the value of y back into one of the original equations. Let's use the first equation:
2x - 3(-13/19) = 5
2x + 39/19 = 5
2x = 5 - 39/19
2x = 95/19 - 39/19
2x = 56/19
6. Solve for x by dividing both sides of the equation by 2:
(2x) / 2 = (56/19) / 2
x = 28/19
Therefore, the solution to the system is x = 28/19 and y = -13/19.
In this case, the addition method was preferred over the substitution method because the coefficients in both equations were simple numbers. The manipulation required to eliminate variables was straightforward and allowed for easier calculations.