Math / Algebra

QuestionSolve the system by the method of your choice. $\left\{\begin{array}{l} x=2 y+2 \\ -3 x+6 y=-6 \end{array}\right.$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. There is exactly one solution. The solution set is $\}$ . (Simplify your answer. Type an ordered pair.) $\square$ B. There are infinitely many solutions. The solution set is $\{(x, y) \mid x=2 y+2\}$ or $\{(x, y) \mid-3 x+6 y=-6\}$ . C. The solution set is $\varnothing$ .

Studdy Solution

STEP 1

1. We are given a system of two linear equations: $ \begin{cases} x = 2y + 2 \\ -3x + 6y = -6 \end{cases} \]
2. The goal is to find the solution(s) to this system, which could be a single solution, infinitely many solutions, or no solution.
3. We can use substitution or elimination to solve the system.

STEP 2

1. Use substitution to solve the system.
2. Simplify and solve for $y$ .
3. Substitute back to find $x$ .
4. Verify the solution and determine the nature of the solution set.

STEP 3

Since the first equation is already solved for $x$ , we can substitute $x = 2y + 2$ into the second equation:
$-3(2y + 2) + 6y = -6$

STEP 4

Distribute and simplify the equation:
$-6y - 6 + 6y = -6$
Combine like terms:
$0 = -6$

STEP 5

The equation $0 = -6$ is a contradiction, which means there is no solution to the system.

STEP 6

Since the simplified equation leads to a contradiction, the system of equations has no solution.
The correct choice is:
C. The solution set is $\varnothing$ .

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Studdy Solution

STEP 1

STEP 2

1. Use substitution to solve the system.
2. Simplify and solve for $y$ .
3. Substitute back to find $x$ .
4. Verify the solution and determine the nature of the solution set.

STEP 3

Since the first equation is already solved for $x$ , we can substitute $x = 2y + 2$ into the second equation:
$-3(2y + 2) + 6y = -6$

STEP 4

Distribute and simplify the equation:
$-6y - 6 + 6y = -6$
Combine like terms:
$0 = -6$

STEP 5

The equation $0 = -6$ is a contradiction, which means there is no solution to the system.

STEP 6

Since the simplified equation leads to a contradiction, the system of equations has no solution.
The correct choice is:
C. The solution set is $\varnothing$ .

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Studdy Solution

STEP 1

STEP 2

1. Use substitution to solve the system.2. Simplify and solve for y y y.3. Substitute back to find x x x.4. Verify the solution and determine the nature of the solution set.

STEP 3

Since the first equation is already solved for x x x, we can substitute x=2y+2 x = 2y + 2 x=2y+2 into the second equation:−3(2y+2)+6y=−6 -3(2y + 2) + 6y = -6 −3(2y+2)+6y=−6

STEP 4

Distribute and simplify the equation:−6y−6+6y=−6 -6y - 6 + 6y = -6 −6y−6+6y=−6Combine like terms:0=−6 0 = -6 0=−6

STEP 5

The equation 0=−6 0 = -6 0=−6 is a contradiction, which means there is no solution to the system.

STEP 6

Since the simplified equation leads to a contradiction, the system of equations has no solution.The correct choice is:C. The solution set is ∅\varnothing∅.

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1. Use substitution to solve the system.
2. Simplify and solve for $y$ .
3. Substitute back to find $x$ .
4. Verify the solution and determine the nature of the solution set.

Since the first equation is already solved for $x$ , we can substitute $x = 2y + 2$ into the second equation:
$-3(2y + 2) + 6y = -6$

Distribute and simplify the equation:
$-6y - 6 + 6y = -6$
Combine like terms:
$0 = -6$

The equation $0 = -6$ is a contradiction, which means there is no solution to the system.

Since the simplified equation leads to a contradiction, the system of equations has no solution.
The correct choice is:
C. The solution set is $\varnothing$ .