Question

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

Answer 1

To solve a nonlinear system of equations using substitution, follow these steps with the appropriate answers filled in:

### Step 1: Substitute
Take the linear equation and solve for one variable in terms of the other. For example, if you have a linear equation like \( y = mx + b \), you can express \( y \) in terms of \( x \).

### Step 2: Substitute into the Nonlinear Equation
Substitute the expression from the linear equation into the nonlinear equation. For example, if your nonlinear equation is \( x^2 + y^2 = r^2 \) (the equation of a circle), substitute the expression for \( y \) from the first step into this equation.

### Step 3: Solve for One Variable
After substituting, you will end up with an equation that only contains one variable (e.g., just \( x \)). Solve this equation for that variable. For instance, from the equation \( x^2 + (mx + b)^2 = r^2 \), you would solve for \( x \).

### Step 4: Substitute Back to Find the Other Variable
Once you have the value(s) for \( x \), substitute those values back into the original linear equation to find the corresponding values of \( y \).

### Step 5: Check Your Solutions
After finding the pairs \((x, y)\), it’s important to verify that these pairs satisfy both equations in the original system. Substitute your solutions into both the linear and the nonlinear equation to ensure they hold true.

### Example:
Let's say we have the following system:

1. Linear equation: \( y = 2x + 3 \)
2. Nonlinear equation: \( x^2 + y^2 = 25 \)

**Step 1:** Substitute \( y = 2x + 3 \) into the nonlinear equation.

**Step 2:** The nonlinear equation becomes \( x^2 + (2x + 3)^2 = 25 \).

**Step 3:** Expand and solve for \( x \):

\[
x^2 + (4x^2 + 12x + 9) = 25
\]
\[
5x^2 + 12x - 16 = 0
\]

(use the quadratic formula or factor to solve for \( x \)).

**Step 4:** Once values of \( x \) are found, substitute them back into \( y = 2x + 3 \) to find corresponding \( y \) values.

**Step 5:** Finally, check the pairs \((x, y)\) by substituting them back into both original equations.

By following these steps systematically, you can successfully solve a nonlinear system of equations using substitution.