Solve the following systems of linear equations using algebraic method.
1)3y=11-2x
3x=y-11
2)8x-6y=14
12x-9y=18
1). 3y=11-2x
2). 3x=y-11
Using equation 1: y = (11-2x)/3
Substituting into equation 2
3x = (11-2x)/3 - 11
x = (11 -2x)/9 - 11/3
x + 2x/9 = 11/9 - 11/3
11x/9 = 11/9 - 11/3
solve for x
Then plug x into either equation 1 or 2, and solve for y
2) Use a similar procedure, as shown in problem 1
To solve the systems of linear equations using the algebraic method, we will use the method of substitution or the method of elimination to find the values of x and y.
1) Solve the system of equations:
3y = 11 - 2x ...(1)
3x = y - 11 ...(2)
We will use the method of substitution:
Step 1: Solve equation (2) for y:
y = 3x + 11 ...(3)
Step 2: Substitute equation (3) into equation (1):
3(3x + 11) = 11 - 2x
Simplify the equation:
9x + 33 = 11 - 2x
Step 3: Move all the x terms to one side and the constant terms to the other side:
9x + 2x = 11 - 33
Simplify the equation:
11x = -22
Step 4: Solve for x by dividing both sides by 11:
x = -2
Step 5: Substitute the value of x into equation (3) to find y:
y = 3(-2) + 11
y = 5
Therefore, the solution to the system of equations is x = -2 and y = 5.
2) Solve the system of equations:
8x - 6y = 14 ...(4)
12x - 9y = 18 ...(5)
We will use the method of elimination:
Step 1: Multiply equation (4) by 3 and equation (5) by 2 to make the coefficients of x in both equations equal:
24x - 18y = 42 ...(6)
24x - 18y = 36 ...(7)
Step 2: Subtract equation (7) from equation (6) to eliminate the x variable:
(24x - 18y) - (24x - 18y) = 42 - 36
Simplify the equation:
0 = 6
Step 3: Since the equation is not true, it means that the system of equations is inconsistent. This means that the system of equations does not have a solution.
Therefore, the system of equations in question 2) does not have a solution.