solve the equations algebraically
x^2 + y^2 = 25
3y - 4x = 0
I would use substitution, from the 2nd ....
y = 4x/3
sub into the first ....
x^2 + (4x/3)^2 = 25
x^2 + 16x^2/9 = 25
9x^2 + 16x^2 = 225
25x^2 = 225
x^2 = 9
x = ± 3
if x=3, y = 4
if x = -3 , y = -4
To solve this system of equations algebraically, we can use the method of substitution.
Let's start by solving the second equation for y in terms of x:
3y - 4x = 0
3y = 4x
y = (4/3)x
Now, substitute this value of y into the first equation:
x^2 + y^2 = 25
x^2 + ((4/3)x)^2 = 25
x^2 + (16/9)x^2 = 25
(1 + (16/9))x^2 = 25
(25/9)x^2 = 25
Next, divide both sides of the equation by (25/9):
x^2 = (25 * 9) / 25
x^2 = 9
Now, take the square root of both sides:
√(x^2) = √9
x = ±3
So, the possible values for x are x = 3 and x = -3.
Substitute these values back into the equation y = (4/3)x to find the corresponding values of y:
For x = 3:
y = (4/3)(3) = 4
For x = -3:
y = (4/3)(-3) = -4
Therefore, the solutions to the system of equations are (x, y) = (3, 4) and (x, y) = (-3, -4).
To solve the system of equations algebraically, we can use the method of substitution or elimination. Let's use the method of substitution to solve these equations.
First, solve one of the equations for one variable in terms of the other variable. In this case, let's solve the second equation for y in terms of x:
3y - 4x = 0
Add 4x to both sides:
3y = 4x
Divide both sides by 3:
y = (4/3)x
Now substitute this expression for y in the first equation:
x^2 + (4/3)x^2 = 25
Multiply both sides by 3 to get rid of the fraction:
3x^2 + 4x^2 = 75
Combine like terms:
7x^2 = 75
Divide both sides by 7:
x^2 = 75/7
Now take the square root of both sides:
x = ± √(75/7)
Simplifying the square root:
x = ± √(75)/√(7)
x = ± √(25*3)/√(7)
x = ± 5√(3)/√(7)
So the x-values are x = 5√(3)/√(7) and x = -5√(3)/√(7).
Finally, substitute these x-values back into the equation y = (4/3)x to find the corresponding y-values:
For x = 5√(3)/√(7):
y = (4/3)(5√(3)/√(7))
Simplifying:
y = (20√(3))/(3√(7))
For x = -5√(3)/√(7):
y = (4/3)(-5√(3)/√(7))
Simplifying:
y = (-20√(3))/(3√(7))
Therefore, the solutions to the system of equations are:
(x, y) = (5√(3)/√(7), 20√(3)/(3√(7))) and (x, y) = (-5√(3)/√(7), -20√(3)/(3√(7)))