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:

\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!