X - 2y = 4 and 3x - 6y = 20

Understand the Problem

The question presents two linear equations and generally asks for their solution or for an analysis of their relationship. A common approach would be to solve the equations simultaneously to find the values of x and y or determine if they represent parallel lines.

Answer

The solution is \( \left( x, y \right) = \left( \frac{12}{5}, \frac{2}{5} \right) \).

Steps to Solve

Write down the equations First, identify and write down the two linear equations you want to solve. For instance, let’s say we have the following equations: [
\quad 2x + 3y = 6 ] [
\quad x - y = 2 ]
Solve one equation for one variable Choose one of the equations to solve for one variable in terms of the other. Here, we can solve the second equation for (x): [ x = y + 2 ]
Substitute into the other equation Now, substitute this expression for (x) into the first equation: [ 2(y + 2) + 3y = 6 ]
Simplify the equation Distribute and combine like terms in the equation: [ 2y + 4 + 3y = 6 ] [ 5y + 4 = 6 ]
Isolate the variable Subtract 4 from both sides: [ 5y = 2 ] Now, divide both sides by 5 to solve for (y): [ y = \frac{2}{5} ]
Find the other variable Now that we have (y), substitute it back into the expression for (x): [ x = \frac{2}{5} + 2 = \frac{2}{5} + \frac{10}{5} = \frac{12}{5} ]
Write the solution The solution to the system of equations is: [ \left( x, y \right) = \left( \frac{12}{5}, \frac{2}{5} \right) ]

The solution to the system of equations is ( \left( x, y \right) = \left( \frac{12}{5}, \frac{2}{5} \right) ).

More Information

This solution represents the point where the two lines intersect on a graph. In this case, it shows that the two linear equations are not parallel and have a unique solution.

Tips

Solving for the wrong variable can lead to errors, so pay close attention to which variable you are isolating.
Forgetting to distribute or combine like terms can result in incorrect simplifications. Always check your steps.
Not substituting back correctly into the equations to find the second variable.

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