UP Board Class 10 Linear Equations in One Variable — practice questions — page 2

A system of two linear equations in two variables, $x$ and $y$, represents two straight lines in a Cartesian plane. The solution to the system corresponds to the point(s) where the two lines share the same $(x, y)$ coordinates. There are three possible geometric relationships between two lines: 1. **Intersecting Lines**: If the lines have different slopes, they cross at exactly one point. This point $(x, y)$ represents the **unique solution**. 2. **Parallel Lines**: If the lines have the same slope but different $y$-intercepts, they never meet. This means the system has **no solution**. 3. **Coincident Lines**: If the lines have the same slope and the same $y$-intercept, they lie exactly on top of each other. This means they share every point, resulting in **infinitely many solutions**. Therefore, a unique solution exists if and only if the lines intersect at exactly one point.

The **elimination method** is a strategy for solving systems of linear equations. The primary objective is to 'eliminate' one of the variables ($x$ or $y$) so that you are left with a single equation containing only one variable. ### How it works: 1. **Standard Form:** Ensure both equations are in the form $Ax + By = C$. 2. **Align Coefficients:** Multiply one or both equations by a constant so that the coefficients of one variable are either the same or opposite. For example, if you have $2x$ in the first equation, you might multiply the second equation to get $-2x$. 3. **Add or Subtract:** - If the coefficients are opposites (e.g., $3y$ and $-3y$), **add** the equations. - If the coefficients are the same (e.g., $5x$ and $5x$), **subtract** the equations. 4. **Solve:** Solve the resulting single-variable equation, then substitute that value back into one of the original equations to find the second variable. **Example:** Given: 1) $x + y = 10$ 2) $x - y = 2$ Adding them eliminates $y$: $(x + x) + (y - y) = 10 + 2$ $2x = 12$ $x = 6$ Since we manipulate the equations through multiplication to create this cancellation, the correct option is **Multiply equations so one variable cancels on add/subtract**.