Solve the system of nonlinear equations. x^2+y^2=5 and y=3x-5

Question

Solve the system of nonlinear equations.

x^2+y^2=5

y=3x-5

Answer 1

To solve the system of nonlinear equations, we can use the substitution method. First, we'll substitute the expression for y from the second equation into the first equation:

x^2 + y^2 = 5 y = 3x - 5

Now, substitute the expression for y in the first equation:

x^2 + (3x - 5)^2 = 5

Expand and simplify the equation:

x^2 + (9x^2 - 30x + 25) = 5

Combine like terms:

10x^2 - 30x + 25 = 5

Subtract 5 from both sides:

10x^2 - 30x + 20 = 0

Divide both sides by 10:

x^2 - 3x + 2 = 0

Factor the quadratic equation:

(x - 1)(x - 2) = 0

This gives us two possible values for x:

x = 1 x = 2

Now, we can find the corresponding values for y using the expression y = 3x - 5:

For x = 1: y = 3(1) - 5 y = 3 - 5 y = -2

For x = 2: y = 3(2) - 5 y = 6 - 5 y = 1

So we have two solutions for the system of nonlinear equations:



(x, y) = (1, -2) and (x, y) = (2, 1)

Answer 2

Question: Solve the system of nonlinear equations.

1. x^2+y^2=5
2. y=3x-5

How to solve the equation: Use the Substitution Method

Step 1: Substitute y in the first equation.

Substitute y = 3x -5 into x^2 + y^2 = 5

x^2 + (3x-5)^2 = 5

Step 2: Expand and simplify the equation

Expand (3x-5)^2

(3x-5) x (3x-5) Use FOIL method

3x times 3x = 9x^2 3x times -5 = -15x

-5 times 3x = -15x - 5 times -5 = 25

= 9x^2-15x-15x+25 = 9x^2-30x +25

Substitute back into the equation:

x^2 + 9x^2 - 30x + 25 = 5

Combine like terms:

10x^2 - 30x + 20 = 0

Step 3:

Use the Quadratic Equation to solve for x:

Divide the equation ( 10x^2 - 30x + 20 = 0) by 10:

x2 - 3x + 2 = 0 Factor the quadratic equation: (x-1) (x-2) = 0

Solve for x:

x-1= 0 add 1 to both sides x = 1

x-2 = 0 add 2 to both sides x = 2

Step 4:

Find the corresponding y values:

Substitute x = 1 into y = 3x - 5

y = 3 (1) - 5 = - 2

y = 3 - 5 = - 2

Substitute x = 2 into y = 3x - 5

y = 3 (2) - 5 = 1

y = 6 - 5 = 1

Step 5:

Find the final solution.

(x1,y1) and (x2,y2)

(1,-2) and (2,1)

The solutions to the system equations is (1,-2) and (2,1)

I hope the mathematical calculations (step by step approach) was helpful. Please let me know if you require any more assistance. If anyone in my neighborhood is interested in setting up an in-person math tutoring session. I look forward to hearing from them. Have an amazing day. Doris H.

Answer 3