NCERT Solutions Class 10 Maths Chapter 3 — Pair of Linear Equations in Two Variables

Solution:

Step 1: A solution to a pair of linear equations must satisfy both equations when the values of x and y are substituted into them.

Step 2: Graphically, this solution represents the point where the two lines corresponding to the equations intersect.

Solution:

Step 1: When two lines intersect at a single point, that point is the unique common point to both lines.

Step 2: Therefore, the system of equations has exactly one solution, also known as a unique solution.

Solution:

Step 1: For a system of linear equations `a1x + b1y + c1 = 0` and `a2x + b2y + c2 = 0` to not have a unique solution, we must have a1/a2 = b1/b2.

Step 2: From the given equations, `a1=1`, `b1=k`, `a2=3`, `b2=2`.

Step 3: So, we set the ratios equal: `1/3 = k/2`.

Step 4: Cross-multiplying gives `2 = 3k`, which means `k = 2/3`.

Solution:

Step 1: Ravi has derived `x = 7 - y` from the first equation.

Step 2: To use the substitution method, this expression for `x` must be substituted into the second equation, `2x - 3y = 4`.

Step 3: Replacing `x` with `(7 - y)` in the second equation gives `2(7 - y) - 3y = 4`.

Solution:

Step 1: The phrase 'their sum is 8' translates directly to the equation `x + y = 8`.

Step 2: The phrase 'twice the first number minus the second number is 7' translates to `2x - y = 7`.

Step 3: Combining these, the correct pair of linear equations is `x + y = 8` and `2x - y = 7`.