Two rails are represented by the equations $x + 2y - 4 = 0$ and $2x + 4y - 12 = 0$ . Will the rails cross each other?

Answer: The rails will not cross each other.

Step-by-step solution

Step 1: Identify the given linear equations

We are given two linear equations representing the two rails. Let's label them as $L_1$ and $L_2$ for clarity. If the rails cross each other, then there must be a common solution $(x, y)$ that satisfies both equations. Otherwise, they are parallel and will never cross.

Step 2: Express x in terms of y from the first equation

To solve this system of linear equations using the substitution method, we first express one variable in terms of the other from one of the equations. From the first equation, $x + 2y - 4 = 0$ , we can isolate $x$ by moving the other terms to the right side of the equation.

Step 3: Substitute the expression for x into the second equation

Now we substitute the expression for $x$ (which is $4 - 2y$ ) from the first equation into the second equation, $2x + 4y - 12 = 0$ . This will give us an equation with only one variable, $y$ .

Step 4: Simplify and solve for y

We expand the term $2(4 - 2y)$ to get $8 - 4y$ . Then, we combine like terms in the equation. We see that the $y$ terms cancel each other out, leading to a numerical statement.

Step 5: Interpret the result

The resulting statement, $-4 = 0$ , is false. This means that there is no value of $y$ that can satisfy both equations simultaneously. When a system of linear equations leads to a false statement after substitution, it indicates that the equations represent parallel lines that never intersect. Therefore, the rails will not cross each other.

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Two rails are represented by the equations x+2y−4=0x + 2y - 4 = 0x+2y−4=0 and 2x+4y−12=02x + 4y - 12 = 02x+4y−12=0. Will the rails cross each other?