The sum of two expressions is x-y-2XY+Y2-7. a If one of them is 2x+34-7 Y2 t 1, find the other?
Answers
Step-by-step explanation:
x^2 + 2xy + y^2 = 25 can be factored into
(x + y)^2 = 25
taking the square root of both sides of the equation
x + y = 5 and x + y = -5
putting these linear equations into slope-intercept form y = mx + b, where m is the slope and b is the y-intercept.
Equation 1
x + y = 5
y = -x + 5 (m = -1 and b = 5)
Equation 2
x + y = -5
y = -x - 5 (m = -1 and b = -5)
Notice: these lines have the same slope m = -1, so they are parallel by definition.
Since the solution set is the intersection of x - y = 7 and these two lines, we set the latter equation in slope intercept form.
Equation 3
x - y = 7
-y = -x + 7
y = x - 7
We can now substitute the value of y from the Equation 3 into Equation 1 to get one solution to this system of equations:
x - 7 = -x + 5
2x = 5 + 7 = 12
x = 6
Now substitute x = 6 into y = -x + 5 and solve for y
y = -x + 5
y = -6 + 5
y = -1
So one solution is the point (6, -1)
To get the second solution, just repeat the process, substituting the value of y from the Equation 3 into Equation 2.
Answer:
Let
2x+3y
1
=u and
3x−2y
1
=v. Then, the given system of equations becomes
2
1
u+
7
12
v=
2
1
⇒7u+24v=7 ..(i)
and, 7u+4v=2 .(ii)
Substracting equation (ii) from equation (i), we get
20v=5⇒v=
4
1
Putting v=
4
1
in equation (i), we get
7u+6=7⇒u=
7
1
Now, u=
7
1
⇒
2x+3y
1
=
7
1
⇒2x+3y=7 (iii)
and, u=
4
1
⇒
3x−2y
1
=
4
1
⇒3x−2y=4 .(iv)
Multiplying equation (iii) by 2 and equation (iv) by 3, we get
4x+6y=14 .(v)
9x−6y=12 .(vi)
Adding equations (v) and (vi), we get
13x=26⇒x=2
Putting x=2 in equation(v), we get
8+6y=14⇒y=1
Hence, x=2,y=1 is the solution of the given system of equations.
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Step-by-step explanation:
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