Question

y = 3x² + 3x - 5
2x - y = 3

Explanation:

Step1: Express y from the second equation

From \(2x - y = 3\), we can rearrange it to get \(y = 2x - 3\).

Step2: Substitute y into the first equation

Substitute \(y = 2x - 3\) into \(y = 3x^2 + 3x - 5\), we have:
\(2x - 3 = 3x^2 + 3x - 5\)

Step3: Rearrange into standard quadratic form

Rearrange the equation: \(3x^2 + 3x - 5 - 2x + 3 = 0\)
Simplify to: \(3x^2 + x - 2 = 0\)

Step4: Solve the quadratic equation

Factor the quadratic equation: \(3x^2 + x - 2 = (3x - 2)(x + 1) = 0\)
Set each factor equal to zero:

• \(3x - 2 = 0\) gives \(x = \frac{2}{3}\)
• \(x + 1 = 0\) gives \(x = -1\)

Step5: Find the corresponding y values

For \(x = \frac{2}{3}\), substitute into \(y = 2x - 3\): \(y = 2\times\frac{2}{3} - 3 = \frac{4}{3} - 3 = -\frac{5}{3}\)
For \(x = -1\), substitute into \(y = 2x - 3\): \(y = 2\times(-1) - 3 = -2 - 3 = -5\)

Answer:

The solutions are \(x = \frac{2}{3}, y = -\frac{5}{3}\) and \(x = -1, y = -5\)