Question

solve the system bysubstitution. y=×^2-2×+1 y=2×+1

Answer 1

[AI-generated answer — Bot GPT 3.5]

To solve the system of equations by substitution, we can substitute one equation into the other equation and solve for the variable.

Substituting the expression for y from the second equation into the first equation, we have:

2x + 1 = x^2 - 2x + 1

Rearranging this equation, we get:

x^2 - 4x = 0

Factoring out an x, we have:

x(x - 4) = 0

This equation is satisfied when x = 0 or x - 4 = 0.

For x = 0, substituting this value back into the second equation, we get:

y = 2(0) + 1 = 1

Therefore, one solution to the system of equations is (0, 1).

For x - 4 = 0, we have:

x = 4

Substituting this value back into the second equation, we get:

y = 2(4) + 1 = 9

Therefore, another solution to the system of equations is (4, 9).

Hence, the system of equations has two solutions: (0, 1) and (4, 9).

Answer 2

[AI-generated answer — Explain Bot]

To solve the system of equations by substitution, we can start by solving one of the equations for one variable and substituting it into the other equation. Let's solve the first equation (y = x^2 - 2x + 1) for y:

y = x^2 - 2x + 1

Now, we can substitute this expression for y in the second equation (y = 2x + 1):

x^2 - 2x + 1 = 2x + 1

Next, we can simplify the equation:

x^2 - 2x - 2x = 0

x^2 - 4x = 0

Now, we can factor out the common factor:

x(x - 4) = 0

From here, we can set each factor equal to zero and solve for x:

x = 0 or x - 4 = 0

Solving for x, we have two possible solutions:

x = 0 or x = 4

To find the corresponding y-values, we substitute these values back into either of the original equations. Let's use the first equation:

For x = 0:
y = 0^2 - 2(0) + 1 = 1

For x = 4:
y = 4^2 - 2(4) + 1 = 9

Therefore, the solution to the system of equations is:
(x, y) = (0, 1) and (4, 9)

Answer 3

[AI-generated answer — Step-by-Step Bot]

To solve the system of equations by substitution, we need to substitute one equation into the other. In this case, we have:

Equation 1: y = x^2 - 2x + 1
Equation 2: y = 2x + 1

We can substitute the expression for y from Equation 2 into Equation 1.

Substituting Equation 2 into Equation 1, we get:

x^2 - 2x + 1 = 2x + 1

Now we have a quadratic equation. We can solve it by bringing all the terms to one side:

x^2 - 2x - 2x + 1 - 1 = 0

Simplifying, we get:

x^2 - 4x = 0

Factoring the equation, we have:

x(x - 4) = 0

Setting each factor equal to zero and solving for x, we get:

x = 0 or x - 4 = 0

which gives us two possible values for x: x = 0 or x = 4.

Now, substituting these values back into either Equation 1 or Equation 2, we can find the corresponding values of y.

For x = 0:
Using Equation 2, y = 2(0) + 1 = 1.

For x = 4:
Using Equation 2, y = 2(4) + 1 = 9.

So the solutions to the system of equations are (0, 1) and (4, 9).