Question
Solve the system of equations. x + 4y = 3 -5x - 20y = -15 (1 point) Responses no solution no solution (0,0) (0,0) infinitely many solutions infinitely many solutions (3,0)
Answers
Answered by
GPT-4o mini
To solve the system of equations:
1. \( x + 4y = 3 \) (Equation 1)
2. \( -5x - 20y = -15 \) (Equation 2)
We can start by simplifying Equation 2. Notice that we can factor out -5 on the left side:
\[
-5(x + 4y) = -15
\]
Dividing both sides by -5 gives us:
\[
x + 4y = 3 \quad \text{(which is the same as Equation 1)}
\]
Since both equations represent the same line, they will have infinitely many solutions. Any point that satisfies the equation \( x + 4y = 3 \) is a solution to the system.
To express the solutions in a different way, we can solve Equation 1 for \( x \):
\[
x = 3 - 4y
\]
Thus, every point of the form \( (3 - 4y, y) \) for any real number \( y \) is a solution.
**Conclusion: The system has infinitely many solutions.**
1. \( x + 4y = 3 \) (Equation 1)
2. \( -5x - 20y = -15 \) (Equation 2)
We can start by simplifying Equation 2. Notice that we can factor out -5 on the left side:
\[
-5(x + 4y) = -15
\]
Dividing both sides by -5 gives us:
\[
x + 4y = 3 \quad \text{(which is the same as Equation 1)}
\]
Since both equations represent the same line, they will have infinitely many solutions. Any point that satisfies the equation \( x + 4y = 3 \) is a solution to the system.
To express the solutions in a different way, we can solve Equation 1 for \( x \):
\[
x = 3 - 4y
\]
Thus, every point of the form \( (3 - 4y, y) \) for any real number \( y \) is a solution.
**Conclusion: The system has infinitely many solutions.**