Solve the given systems of equations by substitution:  
3x – y = -2
2x2 – y = 0


To solve the system of equations by substitution, follow these steps:

### Given System:
1. \( 3x - y = -2 \)
2. \( 2x^2 - y = 0 \)

### Step 1: Solve one equation for one variable
Let's solve the first equation for \( y \):

\[
3x - y = -2
\]

Add \( y \) to both sides:

\[
3x = y - 2
\]

Add 2 to both sides:

\[
y = 3x + 2
\]

### Step 2: Substitute into the second equation
Substitute \( y = 3x + 2 \) into the second equation \( 2x^2 - y = 0 \):

\[
2x^2 - (3x + 2) = 0
\]

Distribute the negative sign:

\[
2x^2 - 3x - 2 = 0
\]

### Step 3: Solve the quadratic equation
To solve the quadratic equation \( 2x^2 - 3x - 2 = 0 \), we can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 2 \), \( b = -3 \), and \( c = -2 \). Plugging in these values:

\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-2)}}{2(2)}
\]

Simplify:

\[
x = \frac{3 \pm \sqrt{9 + 16}}{4}
\]

\[
x = \frac{3 \pm \sqrt{25}}{4}
\]

\[
x = \frac{3 \pm 5}{4}
\]

This gives two possible values for \( x \):

1. \( x = \frac{3 + 5}{4} = \frac{8}{4} = 2 \)
2. \( x = \frac{3 - 5}{4} = \frac{-2}{4} = -\frac{1}{2} \)

### Step 4: Find the corresponding \( y \) values
For each \( x \) value, substitute back into the equation \( y = 3x + 2 \) to find \( y \):

1. When \( x = 2 \):
   \[
   y = 3(2) + 2 = 6 + 2 = 8
   \]
   
2. When \( x = -\frac{1}{2} \):
   \[
   y = 3\left(-\frac{1}{2}\right) + 2 = -\frac{3}{2} + 2 = -\frac{3}{2} + \frac{4}{2} = \frac{1}{2}
   \]

### Solution:
The system of equations has two solutions:
\[
(x, y) = (2, 8) \quad \text{and} \quad \left(-\frac{1}{2}, \frac{1}{2}\right)
\]

Would you like further details or have any questions?

Here are 5 related questions you might find useful:
1. How would you solve the system using the elimination method?
2. Can you graph the system of equations to verify the solutions?
3. What are the steps to solving systems of nonlinear equations in general?
4. How can substitution be used for systems with three variables?
5. What if the quadratic equation didn't factor neatly; how would you proceed?

**Tip:** Always check your solutions by substituting them back into the original equations.

Learn how to solve the given system of equations using the substitution method: 3x – y = -2 and 2x^2 – y = 0. Detailed step-by-step solution provided with examples of solving quadratic equations. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation