Questions: Solve the following system of equations by the substitution method. Check the solutions. y = x^2 y = x + 2

Solution Steps

To solve the system of equations using the substitution method, we will:

1. Substitute \( y = x^2 \) from the first equation into the second equation \( y = x + 2 \).
2. Solve the resulting quadratic equation for \( x \).
3. Substitute the values of \( x \) back into either original equation to find the corresponding \( y \) values.
4. Check the solutions by substituting the values back into both original equations.

We start with the following system of equations: \[ \begin{cases} y = x^2 \quad (1) \\ y = x + 2 \quad (2) \end{cases} \]

We substitute \( y \) from equation (1) into equation (2): \[ x^2 = x + 2 \] Rearranging gives us the quadratic equation: \[ x^2 - x - 2 = 0 \] Factoring this equation, we find: \[ (x + 1)(x - 2) = 0 \] Thus, the solutions for \( x \) are: \[ x = -1 \quad \text{and} \quad x = 2 \]

Next, we substitute the values of \( x \) back into equation (1) to find the corresponding \( y \) values:

1. For \( x = -1 \): \[ y = (-1)^2 = 1 \] So, one solution is \( (-1, 1) \).

2. For \( x = 2 \): \[ y = (2)^2 = 4 \] So, another solution is \( (2, 4) \).

We check both solutions against the original equations:

1. For \( (-1, 1) \):

   • Equation (1): \( 1 = (-1)^2 \) is true.
   • Equation (2): \( 1 = -1 + 2 \) is true.

2. For \( (2, 4) \):

   • Equation (1): \( 4 = (2)^2 \) is true.
   • Equation (2): \( 4 = 2 + 2 \) is true.

Both solutions satisfy the original equations.

Final Answer