Are my answers correct?
Tell whether the system is inconsistent OR dependent.
1.) -7x+6y=-4
14x-12y=8
2.) 3x-y=7
-6x+2y=8
My answers:
1.) inconsistent
2.) Dependent
correct. good job
7x − 3y = 10
14x − 6y = -15
To check whether a system of linear equations is inconsistent or dependent, you need to solve the system of equations using one of the methods, such as substitution, elimination, or matrix methods. Let's go through the steps for each of the given systems and determine the correct answers.
System 1:
-7x + 6y = -4 ...(1)
14x - 12y = 8 ...(2)
To solve this system, you can use the elimination method by multiplying equation (1) by 2 and equation (2) by 1 to make the coefficients of 'x' in both equations equal:
-14x + 12y = -8 ...(3)
14x - 12y = 8 ...(2)
Now, by adding equation (3) and equation (2), we eliminate the 'x' variable:
-14x + 12y + 14x - 12y = -8 + 8
0 = 0
Since the equation 0 = 0 is always true, it means that the two equations of the system are dependent. Therefore, your answer for system 1 is correct: it is dependent.
System 2:
3x - y = 7 ...(4)
-6x + 2y = 8 ...(5)
To solve this system, you can again use the elimination method by multiplying equation (4) by 2 and equation (5) by 3:
6x - 2y = 14 ...(6)
-18x + 6y = 24 ...(7)
By adding equation (6) and equation (7), we eliminate the 'x' variable:
6x - 2y - 18x + 6y = 14 + 24
-12x + 4y = 38
This is a new equation that does not match either of the original equations. The coefficients of the variables 'x' and 'y' are not proportional, which means they cannot be multiples of each other. Thus, the system is inconsistent, and your answer for system 2 is correct: it is inconsistent.
In conclusion:
1.) The system is dependent.
2.) The system is inconsistent.
Please note that the answers provided above are based on the calculations and solving steps explained.