1. What is the solution of the system?
−x+2y = 4
−4x+y = −5
2. How many solutions does the system have?
−2x+y = 3
4y−4 =x
3. How many solutions does the system have?
3x+y= 5
−2y=6x − 10
4. How many solutions does the system have?
−x−2y= −14
4y=−2x − 12
6. Solve the following system using substitution or elimination.
3x+3y= 27
x−3y= −11
7. Solve the following system using substitution or elimination.
2x+3y= 9
x+5y= 8
oh and can u explain how u got your answer
-Thanks
I'll do a couple for you, but I ain't doing your whole assignment...
#1.
−x+2y = 4
−4x+y = −5
The 1st equation says that x = 2y-4. So, use that in the 2nd equation, and you have
-4(2y-4) + y = -5
-8y+16+y = -5
-7y = -21
y = 3
So, x = 2y-4 = 2
The solution is this (2,3)
Check to be sure it works in both equations.
#2.
−2x+y = 3
4y−4 =x
Rearranging things a bit, we have
y = 2x+3
y = 1/4 x + 1
The slopes are different, so the two lines must intersect somewhere. Two lines can only intersect in single point, so one solution.
So, what can you do on the others? When you get an answer, do a check to be sure it's right.
Thanks
Some suggestions:
I you have a system of equations and one of the equations has a variable with 1 as a coefficient
such as −4x+y = −5 in #1, arrange the equation so the variable sits by itself and use substitution.
e.g.
y = 4x - 5
now plug that into −x+2y = 4
-x + 2(4x-5) = 4
-x + 8x - 8 = 4
7x = 14
x = 2
now back into y = 4x-5
y = 4(2) - 5 = 3
I the two equations have the same slope, they are parallel, so they couldn't possibly intersect
but....
it could be that they gave you two equations that are really the same, one having been disguised.
e.g. look at #3
3x+y= 5
−2y=6x − 10
working on the 2nd
-6x - 2y = -10
divide each term by -2
3x + y = 5, which is the same as the first.
In that case there will be an infinite number of solutions, that is, any point that lies on the
line is a solution:
(1,2), (3,-1), (0,5) ....
Look at #6
3x+3y= 27
x−3y= −11
You could use substitution, and follow my steps that I outlined, BUT, notice that you have
-3y and +3y
That is perfectly set up for elimination, since adding the two equations will "eliminate" the y's
4x + 0 = 16
x = 4
now sub that back into either of the given equations, I will go for the firs
3x+3y= 27
12 + 3y = 27
3y = 15
y = 5 , so you have (4,5) as a solution.
did you notice that the first equation could have been simplified ?
3x+3y= 27
x−3y= −11
x +y = 9 , by dividing each term by 3
x - 3y = -11
now by subtracting them, I can eliminate the x's
0 +y - (-3y) = 9 - (-11)
y + 4y = 9+11
4y = 20
y = 5, now go back into one of the first equations, sub in y = 5 to find the x
I could also have used substitution for #6
using x−3y= −11 ------> x = 3y-11
into the other equation:
3x + 3y = 27
3(3y-11) + 3y = 27
9y - 33 + 3y = 27
12y = 60
y = 5
etc
Showing that there is not ONE way to do these, but you will always get the same
answer no matter what method you use.
Hope this helps
1. To find the solution to this system of equations:
−x+2y = 4
−4x+y = −5
You can solve this system using the method of substitution or elimination.
Using substitution, start by solving one equation for one variable and then substituting the expression into the other equation.
From the first equation, solve for x:
x = 2y + 4
Substitute this expression for x in the second equation:
-4(2y + 4) + y = -5
Simplify:
-8y - 16 + y = -5
-7y - 16 = -5
-7y = 11
y = -11/7
Substitute this value back into the first equation to find x:
-x + 2(-11/7) = 4
-x - 22/7 = 4
-x = 32/7
x = -32/7
Therefore, the solution to the system is x = -32/7 and y = -11/7.
2. To find how many solutions the system has:
−2x+y = 3
4y−4 = x
This system can be solved using substitution or elimination methods.
Rearrange the second equation to solve for x:
x = 4y - 4
Substitute this expression for x in the first equation:
-2(4y - 4) + y = 3
Simplify:
-8y + 8 + y = 3
-7y + 8 = 3
-7y = -5
y = 5/7
Substitute this value back into the second equation to find x:
x = 4(5/7) - 4
x = 20/7 - 28/7
x = -8/7
Since we have found specific values for x and y, there is only one solution to this system.
3. To find how many solutions the system has:
3x+y= 5
-2y=6x − 10
This system can be solved using substitution or elimination methods.
Rearrange the second equation to solve for x:
6x = -2y + 10
x = (-2y + 10) / 6
x = (-y + 5) / 3
Substitute this expression for x in the first equation:
3((-y + 5) / 3) + y = 5
Simplify:
-y + 5 + y = 5
5 = 5
Both equations simplify to the same expression, which means they are the same line and intersect at infinitely many points. Therefore, this system has infinitely many solutions.
4. To find how many solutions the system has:
−x−2y= −14
4y=−2x − 12
This system can be solved using substitution or elimination methods.
Rearrange the second equation to solve for x:
2x = -4y - 12
x = (-4y - 12) / 2
x = -2y - 6
Substitute this expression for x in the first equation:
-(-2y - 6) - 2y = -14
Simplify:
2y + 6 - 2y = -14
6 = -14
The resulting equation is contradictory, meaning there is no solution that satisfies both equations simultaneously. Therefore, this system has no solutions.
6. To solve the system using substitution or elimination:
3x+3y= 27
x−3y= −11
Using the elimination method, you can eliminate one variable by adding or subtracting the equations.
Multiply the second equation by 3 to create equal coefficients for y:
3x + 3y = 27
3(x - 3y) = 3(-11)
Simplify:
3x + 3y = 27
3x - 9y = -33
Now subtract the second equation from the first to eliminate x:
(3x + 3y) - (3x - 9y) = 27 - (-33)
3x + 3y - 3x + 9y = 27 + 33
12y = 60
y = 5
Substitute this value back into either of the original equations to find x:
x - 3(5) = -11
x - 15 = -11
x = 4
Therefore, the solution to the system is x = 4 and y = 5.
7. To solve the system using substitution or elimination:
2x + 3y = 9
x + 5y = 8
Using the elimination method, you can eliminate one variable by adding or subtracting the equations.
Multiply the second equation by 2 to create equal coefficients for x:
2(x + 5y) = 2(8)
2x + 10y = 16
Now subtract the second equation from the first to eliminate x:
(2x + 3y) - (2x + 10y) = 9 - 16
2x + 3y - 2x - 10y = 9 - 16
-7y = -7
y = 1
Substitute this value back into either of the original equations to find x:
x + 5(1) = 8
x + 5 = 8
x = 3
Therefore, the solution to the system is x = 3 and y = 1.