Method 1: Substitution
Step 1: Solve one of the equations for one variable in terms of the other.
Let's solve the first equation for x:
2x = 16 - 3y
x = (16 - 3y)/2
Step 2: Substitute this expression for x in the second equation and solve for y.
6(16 - 3y)/2 - 5y = 20
3(16 - 3y) - 5y = 20
48 - 9y - 5y = 20
-14y = -28
y = 2
Step 3: Substitute the value of y into the expression for x to find the value of x.
x = (16 - 3(2))/2
x = (16 - 6)/2
x = 10/2
x = 5
So, the solution to the system of equations is x = 5 and y = 2.
Method 2: Elimination
Step 1: Multiply the first equation by 6 and the second equation by 2 to make the coefficients of x in both equations equal.
12x + 18y = 96
12x - 10y = 40
Step 2: Subtract the second equation from the first equation to eliminate x.
(12x + 18y) - (12x - 10y) = 96 - 40
12x + 18y - 12x + 10y = 56
28y = 56
y = 2
Step 3: Substitute the value of y back into either of the original equations and solve for x.
2x + 3(2) = 16
2x + 6 = 16
2x = 10
x = 5
So, the solution to the system of equations is x = 5 and y = 2.
Method 3: Matrix
Step 1: Write the augmented matrix for the system of equations.
[2 3 16]
[6 -5 20]
Step 2: Use row operations to transform the augmented matrix into reduced row-echelon form.
R2 = R2 - 3R1/2
[2 3 16]
[0 -13 -8]
R2 = -R2/13
[2 3 16]
[0 1 8/13]
R1 = R1 - 3R2
[2 0 16 - 24/13]
[0 1 8/13]
R1 = R1/2
[1 0 8/13 - 12/13]
[0 1 8/13]
Step 3: Read the solution from the reduced row-echelon form.
x = 8/13
y = 8/13
So, the solution to the system of equations is x = 8/13 and y = 8/13.
Solve the system:
2x + 3y = 16
6x - 5y = 20
Using all 3 ways
1 answer