5x=7+4y
20x-16y=28
how many solutions are there for the system of equations?
Infinitely many solutions
No solution.
two solutions or three solutions?
5x=7+4y, so
5x-4y=7. Multiply both sides by 4, and you get:
20x-16y=28. But that's the other equation you were given - so you've actually only got one equation there, not two. And you can feed any value of x into that equation and get a corresponding value of y out.
To determine how many solutions there are for the system of equations, we can solve them simultaneously using the method of elimination or substitution.
First, let's solve the system of equations using the method of elimination:
Given equations:
5x = 7 + 4y ...(Equation 1)
20x - 16y = 28 ...(Equation 2)
Multiply Equation 1 by 4 to make the coefficients of y the same in both equations:
20x = 28 + 16y ...(Equation 3)
Now we have two equations with the same coefficients for y:
Equation 3: 20x = 28 + 16y
Equation 2: 20x - 16y = 28
Since the left sides of both equations are equal, and the right sides are also equal, these two equations represent the same line. Hence, the system of equations is dependent, meaning there are infinitely many solutions.
Therefore, the answer is: Infinitely many solutions.
To determine the number of solutions for the system of equations, we need to check the consistency and independence of the equations.
Let's solve the equations step by step:
1. Rearrange the first equation to isolate the variables:
5x - 4y = 7.
2. Multiply the second equation by 5 to make the coefficients of x in both equations the same:
20x - 16y = 28.
3. Now we have a system of two equations:
5x - 4y = 7,
20x - 16y = 28.
4. We can see that the second equation is a multiple of the first equation (multiplied by 4). This means that the two equations represent the same line. Hence, these equations are dependent, and they have infinitely many solutions.
5. Therefore, the answer to the question is: Infinitely many solutions.
So, there are infinitely many solutions for the given system of equations.