how many solutions does the system of equations have?
2x= -10+6 and x+5y=3
one
two
infinitely many
none
how many solutions does the system of equations have?
y=5x+7and3y-15x=18
one
two
infinitely many
none
how many solutions does the system of equations have?
x-4y=12and5x-20y=60
one
two
infinitely many
none
how many solutions does the system of equations have?
y-5x=-6 and 3y-15x= -12
one
two
infinitely many
none
what is the solutions of the system?
4x-y= -3
6x-y= -8
(5,5.5)
(-2.5,-7)
(-2.5,2.2)
(2.5,-7)
What is the solution of the system?
5x+6y=20
9x+11y=32
(28, -20)
(28,20)
(-28, 20)
(20, -28)
please help. Please and thank you!
To determine how many solutions a system of equations has, we need to solve the system and check the values of the variables.
1. For the system 2x = -10 + 6 and x + 5y = 3:
Simplifying the first equation, we get 2x = -4, which gives x = -2.
Substituting x = -2 into the second equation, we get -2 + 5y = 3.
Solving for y, we get 5y = 5, which gives y = 1.
Therefore, the system has one solution.
2. For the system y = 5x + 7 and 3y - 15x = 18:
Since the first equation is already solved for y, we can substitute it into the second equation.
Substituting y = 5x + 7 into 3(5x + 7) - 15x = 18, we get 15x + 21 - 15x = 18.
Simplifying, we get 21 = 18, which is not true.
Therefore, the system has no solution.
3. For the system x - 4y = 12 and 5x - 20y = 60:
Simplifying the first equation, we get x = 12 + 4y.
Substituting x = 12 + 4y into the second equation, we get 5(12 + 4y) - 20y = 60.
Simplifying, we get 60 + 20y - 20y = 60.
Simplifying further, we get 60 = 60, which is true.
Since the equations are the same, they represent the same line and will intersect infinitely.
Therefore, the system has infinitely many solutions.
4. For the system y - 5x = -6 and 3y - 15x = -12:
Dividing the second equation by 3, we get y - 5x = -4.
Comparing this equation to the first equation, we see that they are equivalent.
Again, the equations represent the same line and will intersect infinitely.
Therefore, the system has infinitely many solutions.
For the last two questions, we need to solve the systems and find the values of x and y.
5. For the system 4x - y = -3 and 6x - y = -8:
Subtracting the second equation from the first equation, we get (4x - y) - (6x - y) = (-3) - (-8).
Simplifying, we get -2x = 5, which gives x = -5/2 or -2.5.
Substituting x = -2.5 into the first equation, we get 4(-2.5) - y = -3.
Simplifying, we get -10 - y = -3 which gives y = -3 + 10 or y = 7.
Therefore, the solution to the system is (-2.5, 7).
6. For the system 5x + 6y = 20 and 9x + 11y = 32:
Multiplying the first equation by 11 and the second equation by 6, we can eliminate y by subtracting the two equations.
Doing so, we get (55x + 66y) - (54x + 66y) = (220) - (192).
Simplifying, we get x = 28.
Substituting x = 28 into the first equation, we get 5(28) + 6y = 20.
Simplifying, we get 140 + 6y = 20, which gives 6y = -120 and y = -20.
Therefore, the solution to the system is (28, -20).
I hope this helps! Let me know if you have any further questions.
To determine the number of solutions for each system of equations, we need to analyze the coefficients of the variables. You can use a method such as substitution, elimination, or graphing to find the solution(s).
Let's go through each system one by one:
1) 2x = -10 + 6 and x + 5y = 3
The first equation simplifies to 2x = -4. If we divide both sides by 2, we get x = -2.
Plugging this value of x into the second equation, we have -2 + 5y = 3. Solving for y, we get y = 1.
So the system has one solution: (-2, 1).
2) y = 5x + 7 and 3y - 15x = 18
We can rearrange the first equation to express y in terms of x: y = 5x + 7.
Substituting this value of y into the second equation, we have 3(5x + 7) - 15x = 18.
Expanding and simplifying, we get 15x + 21 - 15x = 18.
Since the x terms cancel out, we end up with 21 = 18, which is not true.
Therefore, the system has no solution.
3) x - 4y = 12 and 5x - 20y = 60
For the first equation, let's solve for x in terms of y: x = 4y + 12.
Substituting this value of x into the second equation, we have 5(4y + 12) - 20y = 60.
Expanding and simplifying, we get 20y + 60 - 20y = 60.
The y terms cancel out, and we're left with 60 = 60, which is true no matter the value of y.
Therefore, the system has infinitely many solutions.
4) y - 5x = -6 and 3y - 15x = -12
Let's rearrange the first equation to express y in terms of x: y = 5x - 6.
Substituting this value of y into the second equation, we have 3(5x - 6) - 15x = -12.
Expanding and simplifying, we get 15x - 18 - 15x = -12.
The x terms cancel out, and we're left with -18 = -12, which is not true.
Therefore, the system has no solution.
Now, for the last question, we'll solve the system:
5) 4x - y = -3 and 6x - y = -8
Subtracting the second equation from the first equation, we have (4x - y) - (6x - y) = (-3) - (-8).
Simplifying, we get -2x = 5, or x = -2.5.
Plugging this value of x into either of the original equations, we have 4(-2.5) - y = -3.
Simplifying, we get -10 - y = -3, or y = -7.
Therefore, the solution to the system is (-2.5, -7).
I hope this helps! Let me know if you have any further questions.
assuming you meant
2x= -10y+6
x+5y=3
then multiply the 2nd by 2 and you then have
2x = -10y + 6
2x + 10y = 6
you can see that they are the same equation, so there are infinitely many solutions. That is, any point on the first line is also on the second, since they are the same line.
If you do this same kind of manipulation and you get something like
x+2y = 4
x+2y = -3
then it is clear that no number can satisfy both equations. The graphs are parallel lines.
Finally, if you check the slopes and they are different, then clearly the lines must intersect in exactly one point.
recall that Ax+By=C has slope m = -A/B