\( 3x + 2y + 10x - 2y = 16 + 18 \),

["Solving the Equation: 3x + 2y + 10x - 2y = 16 + 18 – A Complete Guide", "Mathematics often presents equations that may appear simple but require careful attention to detail. One such equation is:", "( 3x + 2y + 10x - 2y = 16 + 18 )", "At first glance, this looks like a linear equation involving two variables, ( x ) and ( y ). In this article, we’ll walk you through the step-by-step solution, clarify common mistakes, and explain how to solve similar linear equations efficiently.", "---", "### Understanding the Equation", "Before solving, simplify both sides of the equation:", "Left side:\n( 3x + 10x + 2y - 2y = (3x + 10x) + (2y - 2y) = 13x + 0y = 13x )", "Right side:\n( 16 + 18 = 34 )", "So the simplified equation becomes:\n( 13x = 34 )", "---", "### Solving for ( x )", "To isolate ( x ), divide both sides by 13:", "[\nx = \frac{34}{13}\n]", "This gives the solution for ( x ), which is a constant because the ( y )-terms canceled out. This indicates the equation represents a vertical line on the coordinate plane at ( x = \frac{34}{13} ).", "---", "### Why Does ( y ) Disappear?", "Notice that the ( +2y ) and ( -2y ) cancel each other (( 2y - 2y = 0 )). This cancellation shows that ( y ) can be any real number — the equation is independent of ( y ). Thus, the solution set includes all ( (x, y) ) points where ( x = \frac{34}{13} ) and ( y ) is any real number.", "---", "### Final Answer", "[\n\boxed{x = \frac{34}{13}}\n]\nwith ( y \in \mathbb{R} ) (any real number).", "---", "### Practical Applications of Solving Linear Equations", "Understanding how to simplify and solve equations like ( 3x + 2y + 10x - 2y = 16 + 18 ) is foundational in algebra. Applications include:", "- Modeling real-world relationships (e.g., cost and quantity in business)\n- Finding intercepts and graphing linear functions\n- Preparing for systems of equations and more complex algebraic reasoning", "---", "### Step-by-Step Summary", "1. Combine like terms on both sides.\n2. Simplify: Left side = ( 13x ), Right side = ( 34 ).\n3. Divide by coefficient of ( x ): ( x = \frac{34}{13} ).\n4. Note ( y ) is free to any real value.", "---", "### Conclusion", "While equations with two variables can initially seem difficult, simplification often reveals key insights—like dependent variables or unique solutions. Remember: if variable terms cancel out in an equation, the solution includes a free variable, making the final result fully dependent on that value alone.", "For further practice, try simplifying other mixed equations or explore systems involving multiple linear equations. Mastery of these basics strengthens advanced math skills and enables better problem-solving across STEM fields.", "---", "Key Terms: linear equation, solve for x, cancel like terms, two variables, algebraic simplification, y independent variable, real number solutions.", "---", "Keywords for SEO:\nsolve 3x + 2y + 10x - 2y = 16 + 18, linear equation simplification, x = 34/13 solution, y cancels out equation, how to solve mixed variable equations, algebra step-by-step guide"]