Math  /  Algebra

QuestionSolve the system by the method of your choice. {x=8y+42x16y=8\left\{\begin{array}{l} x=8 y+4 \\ 2 x-16 y=8 \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.) B. There are infinitely many solutions. The solution set is {(x,y)x=8y+4}\{(x, y) \mid x=8 y+4\} or {(x,y)2x16y=8}\{(x, y) \mid 2 x-16 y=8\}. C. The solution set is \varnothing.

Studdy Solution

STEP 1

1. We are given a system of two linear equations with two variables.
2. We can use substitution or elimination to solve the system.
3. The goal is to find the values of x x and y y that satisfy both equations simultaneously.

STEP 2

1. Substitute the expression for x x from the first equation into the second equation.
2. Solve for y y .
3. Substitute the value of y y back into the first equation to find x x .
4. Verify the solution by checking both equations.
5. Determine the nature of the solution set (one solution, infinitely many solutions, or no solution).

STEP 3

Substitute the expression for x x from the first equation x=8y+4 x = 8y + 4 into the second equation 2x16y=8 2x - 16y = 8 .
2(8y+4)16y=8 2(8y + 4) - 16y = 8

STEP 4

Distribute and simplify the equation:
16y+816y=8 16y + 8 - 16y = 8

STEP 5

Combine like terms:
8=8 8 = 8
This simplifies to a true statement, indicating that the equations are dependent.

STEP 6

Since the simplified equation 8=8 8 = 8 is always true, the system has infinitely many solutions.
The solution set is described by the equation from the first line:
{(x,y)x=8y+4} \{(x, y) \mid x = 8y + 4\}
The correct choice is: B. There are infinitely many solutions. The solution set is {(x,y)x=8y+4}\{(x, y) \mid x = 8y + 4\}.

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