91Ó°ÊÓ

Variable substitution is a powerful technique in solving equations. It involves replacing a variable with its equivalent value or expression. In our problem, we had two variables: the amount of gasoline the small container can hold (\(x\)) and the amount in the big container (=y)). The problem gave us a relationship: the total gasoline is fifty gallons and the big container holds ten gallons less than twice the small container.

By substituting the expression for =y=2x−10) into the equation =x+y=50), we simplified our complex system into a single equation: =x+(2x−10)=50.). This is key in simplifying and solving the problem more efficiently. Variable substitution helps to break down complex relationships step-by-step, making it easier to manage and solve. It is useful in linear equations, quadratic equations, and many other algebra problems.

Linear equations are mathematical statements that show the equality of two expressions with one or more variables. These equations form straight lines when plotted on a graph. In our example, the problem provides us with two linear equations:

ewline=x+y=50
±ð·É±ô¾±²Ô±ð=²â=2³æ−10

To solve these equations, we looked for the intersection point of the lines represented by these equations, but here we used algebraic methods directly. The equations were manipulated and combined to isolate variables and find their values. Solving linear equations often involves steps like:

• Combining like terms
• Isolating a variable
• Using substitution

These are fundamental tools in algebra critical for solving real-world problems just like in our exercise.

Solving problems involving systems of equations requires a clear approach. In this exercise, we followed several well-defined steps:

• Define the Variables: Represent the unknowns with variables (=x) and =y)).
  
• Set Up the Equations: From the problem's information, create your system of equations.
  
• Substitute and Simplify: Use variable substitution to replace one variable with an equivalent expression.
  
• Solve for the Variable: Combine like terms and solve for your variable.
  
• Find the Remaining Variable: Use your obtained value to find the other variable.
  
• Verify the Solution: Check both equations to ensure the solutions satisfy the original problem conditions.
  

Following these steps systematically ensures we don't miss key details and leads us to the correct solution efficiently. This structured approach applies broadly across different math problems and is useful for developing good problem-solving habits.