Solve the system
x^2+y^2-4x+2y=20
4x +3y =5
The first one is a circle centred at (2,-1), or
(x-2)^2+(y+1)^2=20+4+1=25
or
C: (x-2)^2+(y+1)^2=5^2
The second equation is a straight line.
L: 4x+3y=5
To solve the system, set
y=(5-4x)/3
and substitute into equation C
or
(x-2)^2+((5-4x)/3+1)^2=25
Solve for x to get x=5 or x=-1
Substitute into y=(5-4x)/3
to get
y=-5, y=3
So the intersections are
(5,-5), or (-1,3)
To solve the system of equations, we can use the method of substitution or elimination. Let's use the method of substitution.
Step 1: Solve one equation for one variable in terms of the other variable.
Let's solve the second equation for x:
4x + 3y = 5
4x = 5 - 3y
x = (5 - 3y)/4
Step 2: Substitute the expression for x into the other equation.
Substitute x in the first equation:
x^2 + y^2 - 4x + 2y = 20
[(5 - 3y)/4]^2 + y^2 - 4[(5 - 3y)/4] + 2y = 20
Step 3: Simplify and solve the resulting equation.
Expand and simplify the equation:
(25 - 30y + 9y^2)/16 + y^2 - (20 - 12y)/4 + 2y = 20
Multiply every term by 16 to eliminate the denominator:
25 - 30y + 9y^2 + 16y^2 - 80 + 12y + 32y = 320
Combine like terms:
25y^2 - 6y - 95 = 0
Step 4: Solve the quadratic equation.
To solve the quadratic equation, we can factor it or use the quadratic formula. In this case, factoring may not be straightforward, so we will use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
Using a = 25, b = -6, and c = -95, we substitute those values into the formula:
y = (-(-6) ± √((-6)^2 - 4(25)(-95))) / (2(25))
y = (6 ± √(36 + 9500)) / 50
y = (6 ± √(9536)) / 50
y = (6 ± 98) / 50
This gives us two possible values for y:
y = 104/50 or y = -92/50
Simplifying the fractions:
y = 52/25 or y = -46/25
Step 5: Find the corresponding values of x.
Substitute the values of y back into the equation x = (5 - 3y)/4 to solve for x:
For y = 52/25:
x = (5 - 3(52/25)) / 4
x = (5 - 156/25) / 4
x = (125/25 - 156/25) / 4
x = (-31/25) / 4
x = -31/100
For y = -46/25:
x = (5 - 3(-46/25)) / 4
x = (5 + 138/25) / 4
x = (125/25 + 138/25) / 4
x = (263/25) / 4
x = 263/100
So the solutions to the system of equations are:
x = -31/100, y = 52/25
x = 263/100, y = -46/25