How to solve an equation with 2 different variables 

To solve an equation with two different variables, such as \( x \) and \( y \), you typically need two separate equations involving the same variables. This method is called solving a **system of equations**. There are three primary ways to solve such systems:

### 1. **Substitution Method**
This involves solving one of the equations for one variable in terms of the other, then substituting that expression into the second equation. Here are the steps:

- Solve one equation for one variable, e.g., \( x = 2y + 3 \).
- Substitute this expression for \( x \) in the second equation.
- Solve the resulting equation for the remaining variable \( y \).
- Substitute \( y \)'s value back into the first equation to find \( x \).

### Example:
\[
\begin{aligned}
\text{Equation 1:} & \quad x + 2y = 7 \\
\text{Equation 2:} & \quad 3x - y = 8
\end{aligned}
\]
- Solve the first equation for \( x \): \( x = 7 - 2y \).
- Substitute \( x = 7 - 2y \) into the second equation:
  \[
  3(7 - 2y) - y = 8
  \]
- Simplify and solve for \( y \).

### 2. **Elimination Method**
This method involves adding or subtracting the two equations in such a way that one of the variables gets eliminated. Steps:

- Multiply one or both equations by constants so that the coefficients of one variable are opposites.
- Add or subtract the equations to eliminate one variable.
- Solve the remaining equation for the other variable.
- Substitute that value into one of the original equations to solve for the other variable.

### Example:
\[
\begin{aligned}
\text{Equation 1:} & \quad 2x + y = 5 \\
\text{Equation 2:} & \quad 3x - y = 4
\end{aligned}
\]
- Add the equations to eliminate \( y \):
  \[
  (2x + y) + (3x - y) = 5 + 4
  \]
- Solve for \( x \), then substitute back to find \( y \).

### 3. **Graphing Method**
Here, you graph both equations on the same coordinate plane, and the point where the two lines intersect is the solution (the values of \( x \) and \( y \)).

### Example:
If the system is:
\[
y = 2x + 3 \quad \text{and} \quad y = -x + 5,
\]
- Graph each equation, and the point where the two lines intersect will give the solution for \( x \) and \( y \).

### Special Cases:
- **No Solution**: If the lines are parallel, there is no solution (inconsistent system).
- **Infinite Solutions**: If the equations represent the same line, there are infinitely many solutions.

Would you like more details or examples? 

Here are 5 related questions:
1. How do you solve a system of equations with three variables?
2. Can you explain how to determine if a system of equations has no solution or infinitely many solutions?
3. What is a real-life example of solving two-variable systems?
4. How does solving nonlinear systems differ from linear ones?
5. How do you use matrix methods to solve systems of equations?

**Tip**: Always check your solutions by substituting the values back into the original equations!

How to solve an equation with 2 different variables

Learn how to solve a system of equations with two variables using substitution, elimination, and graphing methods. This step-by-step guide covers key algebraic techniques and is suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation