Can you show me an example of solving a system of equations with two variables

Certainly! Let's solve a system of equations with two variables using the substitution method and the elimination method. Here's an example:

**Equations:**
1. \( 2x + y = 10 \)
2. \( x - y = 2 \)

### Substitution Method

1. Solve one of the equations for one variable. Let's solve the second equation for \( x \):

\[ x = y + 2 \]

2. Substitute this expression for \( x \) into the first equation:

\[ 2(y + 2) + y = 10 \]

3. Simplify and solve for \( y \):

\[ 2y + 4 + y = 10 \]

\[ 3y + 4 = 10 \]

\[ 3y = 6 \]

\[ y = 2 \]

4. Substitute \( y = 2 \) back into the equation \( x = y + 2 \):

\[ x = 2 + 2 \]

\[ x = 4 \]

So, the solution is \( x = 4 \) and \( y = 2 \).

### Elimination Method

1. Align the equations:

\[ 2x + y = 10 \]
\[ x - y = 2 \]

2. Add the two equations to eliminate \( y \):

\[ (2x + y) + (x - y) = 10 + 2 \]

\[ 3x = 12 \]

\[ x = 4 \]

3. Substitute \( x = 4 \) back into the second equation \( x - y = 2 \):

\[ 4 - y = 2 \]

\[ y = 2 \]

So, the solution is \( x = 4 \) and \( y = 2 \).

Would you like more details or have any questions? Here are some questions you might want to ask next:

1. Can you explain how to solve a quadratic equation using the quadratic formula?
2. How do you find the intersection point of two lines graphically?
3. Can you solve a system of three equations with three variables?
4. What is the process for solving linear inequalities?
5. How do you determine if a system of equations has no solution or infinitely many solutions?

**Tip:** When using the elimination method, ensure that the coefficients of one variable are the same (or opposites) before adding or subtracting the equations to eliminate that variable.

Explore how to solve a system of equations with two variables using the substitution method and the elimination method. Step-by-step examples and detailed explanations provided. Suitable for Grades 7-9.

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