What are the first few steps to solving this system by the substitution method?
xy=3
x^2+y^2=10
Substitute 3/x for y in the second equation, and solve the resulting equation for x.
x^2 + 9/x^2 = 10
x^4 - 10x^2 + 9 = 0
(x^2 - 1)(x^2 - 9) =
x = 1, -1, 3 or -3.
To solve this system of equations using the substitution method, you can follow these steps:
Step 1: Solve one equation for one variable in terms of the other variable. Let's solve the first equation, xy = 3, for x:
x = 3/y
Step 2: Substitute the expression for x from Step 1 into the other equation. Substitute x = 3/y into the second equation, x^2 + y^2 = 10:
(3/y)^2 + y^2 = 10
Step 3: Simplify the resulting equation by expanding and collecting like terms. In this case, you will need to square both the numerator and denominator:
9/y^2 + y^2 = 10
Step 4: Multiply both sides of the equation by y^2 to eliminate the denominator:
9 + y^4 = 10y^2
Step 5: Rearrange the equation to form a quadratic equation:
y^4 - 10y^2 + 9 = 0
Step 6: Solve the quadratic equation either by factoring, completing the square, or using the quadratic formula. In this case, you can factor the equation as a quadratic in y^2:
(y^2 - 1)(y^2 - 9) = 0
This gives two possible values for y: y^2 = 1 or y^2 = 9.
Step 7: Solve for the corresponding values of x by substituting the values of y back into either of the original equations. Given y^2 = 1, we have two cases:
- For y^2 = 1, y = ±1. Substituting y = 1 into the first equation, we get x = 3/1 = 3. Substituting y = -1 into the first equation, we get x = 3/(-1) = -3.
- For y^2 = 9, y = ±3. Substituting y = 3 into the first equation, we get x = 3/3 = 1. Substituting y = -3 into the first equation, we get x = 3/(-3) = -1.
Step 8: Write the solution as an ordered pair (x, y). The solution to the system of equations is:
(x, y) = {(3, 1), (-3, 1), (1, 3), (-1, 3)}
These are the steps to solve the given system of equations using the substitution method.