Math  /  Algebra

Questiontomework 7.1.39 HW Score: 30%,330 \%, 3 of 10 points Points: 0 of 1
Solve the system. State whether the system is inconsistent or has infinitely many solutions. If the system has infinitely many solutions, write the solution set with yy arbitrary. 2x6y=07x+21y=17\begin{aligned} 2 x-6 y & =0 \\ -7 x+21 y & =17 \end{aligned}
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. There is one solution. The solution set is \square \}. (Simplify your answer. Type an ordered pair.) B. The system has infinitely many solutions. The solution set is \square \}. (Simplify your answer. Type an ordered pair. Type an expression using yy as the variable.) C. The solution is the empty set, \varnothing.

Studdy Solution

STEP 1

1. We are given a system of two linear equations.
2. We need to determine if the system has one solution, infinitely many solutions, or no solution (inconsistent).
3. If there are infinitely many solutions, we need to express the solution set with y y as the arbitrary variable.

STEP 2

1. Analyze the system of equations.
2. Determine if the system is consistent or inconsistent.
3. Solve the system if it is consistent.
4. Express the solution set, if applicable.

STEP 3

Analyze the given system of equations:
2x6y=0(Equation 1)7x+21y=17(Equation 2)\begin{aligned} 2x - 6y &= 0 \quad \text{(Equation 1)} \\ -7x + 21y &= 17 \quad \text{(Equation 2)} \end{aligned}
Notice that both equations have coefficients that are multiples of each other. This suggests that the equations might be dependent or inconsistent.

STEP 4

Determine if the system is consistent or inconsistent by checking if the equations are multiples of each other:
Multiply Equation 1 by 3.5-3.5 to compare with Equation 2:
3.5(2x6y)=3.50-3.5(2x - 6y) = -3.5 \cdot 0
This results in:
7x+21y=0-7x + 21y = 0
Compare this with Equation 2:
7x+21y=17-7x + 21y = 17
Since the left-hand sides are identical but the right-hand sides differ, the system is inconsistent.

STEP 5

Since the system is inconsistent, there are no solutions.
The solution is the empty set, \varnothing.

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