CBSE Class 10 Mathematics Linear Equations Assignment Set 09

X - Mathematics Assignment No. 13 - Linear Equations

Fill the gap.

Question 1. Two linear equations in the same two variables are called ________ in ________ variables
Answer: Pair of linear equations; two
In simple words: Two linear equations with the same pair of variables form a pair of linear equations.

Exam Tip: Remember that the number of variables must remain constant across both equations.

Question 2. The general form of a pair of linear equation is ________ and ________
Answer: \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\)
In simple words: The standard format uses two equations with coefficients for each variable and a constant term.

Exam Tip: Ensure all terms are brought to one side when writing the general form equal to zero.

Question 3. A pair of linear equations in two variables can be solved by ________ and ________
Answer: Graphical and Algebraic method
In simple words: You can find solutions either by plotting lines on a graph or by using algebraic formulas.

Exam Tip: Be familiar with both approaches as exam questions may specify the method to use.

Question 4. If two lines intersect at a point then that point gives the ________ solution, and the equations are called ________ Equations.
Answer: Unique; Consistent
In simple words: A single intersection point means there is only one exact answer, making the system consistent.

Exam Tip: Intersecting lines correspond to the condition \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\).

Question 5. If two lines are parallel then the pair of equations have ________ solution, and the equations are called ________ equations.
Answer: No; Inconsistent
In simple words: Parallel lines never meet, so no solution exists, and the equations are called inconsistent.

Exam Tip: Parallel lines correspond to the condition \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\).

Question 6. If two lines coincides then the have ________ solutions and such type of equations are called ________ equations.
Answer: Infinite; Dependent
In simple words: Overlapping lines share every single point, yielding infinitely many solutions.

Exam Tip: Coincident lines satisfy the ratio condition \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\).

Question 7. In how many ways can a pair of linear equations be solved by algebraic method ________
Answer: Three
In simple words: There are three standard algebraic techniques taught in school for solving linear systems.

Exam Tip: Memorize the three algebraic techniques for quick recall.

Question 8. The methods of solving simultaneous linear equations by algebraic method are ________; ________ and ________
Answer: Substitution, Elimination, Cross multiplication
In simple words: These three methods are substitution, elimination, and cross multiplication.

Question 9. The condition for the linear equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) are for
(i) Consistent equations ________
(ii) Inconsistent equations ________
(iii) Dependant equations ________
Answer:
(i) \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\)
(ii) \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\)
(iii) \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)
In simple words: Ratio comparisons tell you immediately whether lines intersect, stay parallel, or overlap.

Exam Tip: Always write out the coefficient ratios carefully before determining consistency.

Question 10. An equation involving only linear polynomials is called ________
Answer: Linear equation
In simple words: An equation where variables have a highest power of 1 is a linear equation.

Exam Tip: The degree of every term containing a variable must be 1.

Question 11. An equation is a statement of equality of ________ involving an unknown quantity
Answer: Two algebraic expression
In simple words: An equation equates two algebraic expressions containing unknown variables.

Exam Tip: The equals sign (=) is the defining feature of an equation.

Question 12. Two equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) will have ________ if \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)
Answer: Infinite solutions
In simple words: When all three ratios are equal, the equations represent the exact same line.

Exam Tip: Infinite solutions mean dependent and consistent equations.

Question 13. Two lines \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) will be intersecting if ________
Answer: \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\)
In simple words: Non-equal initial ratios guarantee that the lines cross at a single point.

Exam Tip: The constant term \(c_1\) and \(c_2\) do not affect whether lines intersect.

Question 14. Name two lines which are intersecting (write any two equations of your choice) ________ and ________
Answer: \(x + y = 4\) and \(x - y = 1\) (or any other valid intersecting pair)
In simple words: Any two linear equations with different slope ratios will serve as an intersecting pair.

Exam Tip: Simple equations with coefficients that do not form proportions are easiest to write.

Question 15. If two lines are parallel then they are called ________ lines and the condition for such lines are ________
Answer: Inconsistent lines; \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\)
In simple words: Parallel lines never intersect and have no solution, making them inconsistent.

Exam Tip: Verify all three ratios to correctly distinguish parallel lines from coincident lines.