The line 2x+3y=1 intersects the curve x(x+y)=10 at A and B. Calculate the coordinates of A and B.
solve the first for y
3y = 1-2x
y = (1-2x)/3 , now sub that into the other equation
x(x+(1-2x)/3) = 10
x(3x+1-2x)/3 = 10
x(x+1)) = 30
x^2 + x - 30 = 0
(x+6)(x-5) = 0
x = -6 or x = 5
if x = 5, y = (1-10)/3 = -3
if x = -6 , y = (1 + 12)/3 = 13/3
So A = (5,-3) and B is (-6, 13/3)
(of course it doesn't matter who is called A or B)
your mom
To find the coordinates of points A and B, we need to solve the system of equations formed by the line 2x + 3y = 1 and the curve x(x + y) = 10.
Step 1: Solve either equation for one variable in terms of the other.
Let's solve the curve equation, x(x + y) = 10, for x:
x(x + y) = 10
x^2 + xy = 10
x^2 = 10 - xy
x = √(10 - xy)
Step 2: Substitute this expression for x into the equation of the line 2x + 3y = 1.
2(√(10 - xy)) + 3y = 1
Step 3: Simplify and solve for y.
4(10 - xy) + 9y^2 = 1
40 - 4xy + 9y^2 - 1 = 0
9y^2 - 4xy + 39 = 0
Step 4: Apply the quadratic formula to find the values of y.
The quadratic formula is given by:
y = (-b ± √(b^2 - 4ac)) / 2a
Here, a = 9, b = -4x, and c = 39.
Substituting these values into the quadratic formula, we get:
y = (-(-4x) ± √((-4x)^2 - 4*9*39)) / (2*9)
y = (4x ± √(16x^2 - 4*9*39)) / 18
y = (4x ± √(16x^2 - 4*3*13^2)) / 18
y = (4x ± √(16x^2 - 1872)) / 18
Step 5: Substitute the value of y back into the equation x(x + y) = 10 to find the corresponding values of x.
We have two possible values for y, so we will get two corresponding values of x for each.
For the first value of y:
x(x + y) = 10
x(x + (4x + √(16x^2 - 1872)) / 18) = 10
x(18x + 4x + √(16x^2 - 1872)) = 180
22x^2 + 18x^2 + 4x^2 + 4x√(16x^2 - 1872) = 180
For the second value of y:
x(x + y) = 10
x(x + (4x - √(16x^2 - 1872)) / 18) = 10
x(18x + 4x - √(16x^2 - 1872)) = 180
22x^2 + 18x^2 - 4x^2 - 4x√(16x^2 - 1872) = 180
Step 6: Solve the resulting quadratic equations to find the values of x.
To find the values of x, we need to solve the quadratic equations obtained in step 5.
After solving these equations, we will have the corresponding values of x and y for points A and B.
To calculate the coordinates of points A and B, we need to find the values of x and y that satisfy both equations: 2x+3y=1 and x(x+y)=10.
Step 1: Solve the first equation (2x+3y=1) for x in terms of y:
2x = 1 - 3y
x = (1 - 3y)/2
Step 2: Substitute this value of x in the second equation, x(x+y)=10:
(1 - 3y)/2 * ((1 - 3y)/2 + y) = 10 [Substituting x = (1 - 3y)/2]
(1 - 3y)/2 * (1 - 3y + 2y)/2 = 10
(1 - 3y)(1 - y)/4 = 10
Step 3: Expand and simplify the equation:
(1 - 3y)(1 - y) = 40
1 - y - 3y + 3y^2 = 40
3y^2 - 4y - 39 = 0
Step 4: Solve the quadratic equation. We can use factoring, completing the square, or the quadratic formula. Let's use the quadratic formula in this case:
y = (-b ± √(b^2 - 4ac)) / (2a)
Using the values from the quadratic equation (a = 3, b = -4, c = -39), we can plug them into the formula:
y = (-(-4) ± √((-4)^2 - 4 * 3 * -39)) / (2 * 3)
y = (4 ± √(16 + 468)) / 6
y = (4 ± √484) / 6
y = (4 ± 22) / 6
So, we have two possible values for y:
1. y = (4 + 22) / 6 = 26 / 6 = 13/3
2. y = (4 - 22) / 6 = -18 / 6 = -3
Step 5: Substitute the values of y back into the first equation to find the corresponding x values:
For y = 13/3:
x = (1 - 3(13/3)) / 2
x = (1 - 13) / 2
x = -12 / 2
x = -6
So, one solution is A(-6, 13/3).
For y = -3:
x = (1 - 3(-3)) / 2
x = (1 + 9) / 2
x = 10 / 2
x = 5
Another solution is B(5, -3).
Therefore, the coordinates of points A and B are A(-6, 13/3) and B(5, -3), respectively.