Question

How to Use Substitution to Solve a Nonlinear System of Equations
Step 1: Substitute
Answer
Answer
from the linear equation for
in the nonlinear equation.
Step 2: Solve for
Answer
1°
Step 3:
Answer
your solutions into the
1 Answer
equation
and solve for y.
Step 4: Check your solutions.
i Answer,
all solutions into every equation in the system
and verify they are true.
OR
• Answer
is a point of intersection.
all the equations and verify that every solution
:: mx + b
::linear
::Substitute
# Graph
X
Y
# Substitute

Answer 1

To solve a nonlinear system of equations using substitution, follow these detailed steps:

### Step 1: Substitute
Identify a linear equation in the system, typically of the form \( y = mx + b \) or \( x = my + b \). Solve this equation for one variable (e.g., \( y \)) in terms of the other variable (e.g., \( x \)).

**Example:**
Given the linear equation \( y = 2x + 1 \), we identify \( y \) as expressed in terms of \( x \).

### Step 2: Solve for the Other Variable
Take the expression you found from the linear equation and substitute it into the nonlinear equation (which might be quadratic, cubic, etc.).

**Example:**
If the nonlinear equation is \( x^2 + y^2 = 10 \), replace \( y \) with \( 2x + 1 \):

\[ x^2 + (2x + 1)^2 = 10 \]

### Step 3: Solve the Resulting Equation
Now, solve the resulting equation for the remaining variable (in our example, \( x \)).

**Example:**

Expanding \( (2x + 1)^2 \):
\[ x^2 + (4x^2 + 4x + 1) = 10 \]
\[ 5x^2 + 4x + 1 - 10 = 0 \]
\[ 5x^2 + 4x - 9 = 0 \]

Now, solve the quadratic equation using the quadratic formula or factoring, if possible.

### Step 4: Back Substitute
Once you determine the values of \( x \), substitute them back into the original linear equation to find the corresponding \( y \)-values.

**Example:**
Let's say you find \( x = 1 \). Substitute it back:

\[ y = 2(1) + 1 = 3 \]

### Step 5: Check Your Solutions
Verify that your solutions satisfy both the original linear and nonlinear equations.

**Example:**
For the point \( (1, 3) \):
- Check in the linear equation: \( y = 2(1) + 1 \) results in \( 3 = 3 \) (true).
- Check in the nonlinear equation: \( 1^2 + 3^2 = 1 + 9 = 10 \) (true).

### Conclusion
If both conditions hold true, then \( (1, 3) \) is a solution to the system. Repeat this for all solutions derived from the quadratic or nonlinear equation to find all points of intersection.

### Final Notes
- If the systems are more complex, consider graphing both equations to visually confirm intersections.
- Always format your final answers as ordered pairs in the form \( (x, y) \) with both being the solutions to the equations.