Solve the equation 3x+y=25,xy=8

xy = 8, so y = 8/x

3x+y = 25
3x + 8/x = 25
3x^2 - 25x + 8 = 0
now just solve as usual. Watch for extraneous solutions, if any

Yes

Kansozn

Like the idea

To solve the given system of equations:

1. Start by substituting the value of xy from the second equation into the first equation:
3x + y = 25 ⟶ 3x + 8/x = 25

2. Multiply both sides of the equation by x to eliminate the denominator:
3x^2 + 8 = 25x

3. Rearrange the equation to create a quadratic equation in standard form:
3x^2 - 25x + 8 = 0

4. Factor the quadratic equation or use the quadratic formula to find the value(s) of x that satisfy the equation. In this case, the factors are:
(3x - 1)(x - 8) = 0

Setting each factor equal to zero and solving for x results in two values:
3x - 1 = 0 ⟶ 3x = 1 ⟶ x = 1/3

x - 8 = 0 ⟶ x = 8

5. Now that we have the values of x, substitute them back into one of the original equations to find the corresponding y-values:
For x = 1/3:
3(1/3) + y = 25
1 + y = 25
y = 25 - 1
y = 24

For x = 8:
3(8) + y = 25
24 + y = 25
y = 25 - 24
y = 1

Therefore, the solutions to the given system of equations are x = 1/3, y = 24 and x = 8, y = 1.