How can we solve the system of equations: x² + y = 3 and x + y = 3?

Solving the System of Equations

To solve the system of equations:

• Equation 1: x² + y = 3
• Equation 2: x + y = 3

we can use substitution or elimination methods. Here, we will use substitution for simplicity.

Step 1: Solve Equation 2 for y

From Equation 2, we can express y in terms of x:

y = 3 - x

Step 2: Substitute y in Equation 1

Now, substitute this expression for y into Equation 1:

x² + (3 - x) = 3

Now, simplify the equation:

x² + 3 - x = 3

x² - x + 3 - 3 = 0

x² - x = 0

Step 3: Factor the equation

Factor the left-hand side:

x(x - 1) = 0

Setting each factor to zero gives us:

• x = 0
• x = 1

Step 4: Solve for y using the x values

Now, substitute these x values back into Equation 2 to find the corresponding y values:

When x = 0:

y = 3 - 0 = 3

When x = 1:

y = 3 - 1 = 2

Step 5: Present the solutions

The solutions to the system of equations are:

• Solution 1: (0, 3)
• Solution 2: (1, 2)

Conclusion

Therefore, the system of equations x² + y = 3 and x + y = 3 has two solutions: (0, 3) and (1, 2).