Practice Questions

Solve the pair of equations $x + y = 7$ and $x - y = 1$.

The sum of two numbers is 35 and their difference is 13. Formulate a pair of linear equations to represent this situation.

Explain the difference between a consistent and an inconsistent pair of linear equations using their graphical representations.

Define a consistent pair of linear equations.

Formulate a system of linear equations for the following scenario and then solve it: The sum of the present ages of a father and his son is 60 years. Six years ago, the father's age was five times the age of the son.

What is the graphical representation of a pair of linear equations that has no solution?

Analyze the pair of linear equations $x + 2y = 4$ and $2x + 4y = 12$ to determine if their graphical representation will be intersecting lines, parallel lines, or coincident lines without drawing the graph.

Formulate a linear equation in two variables that is parallel to the line $2x - 5y + 7 = 0$ and passes through the point $(5, 3)$.

Name two algebraic methods for solving a pair of linear equations.

State the condition for a pair of linear equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ to have a unique solution.

Analyze if the pair of linear equations $2x - 3y = 7$ and $4x - 6y = 15$ is consistent or inconsistent.

Examine if $x=2, y=1$ is a solution for the system of linear equations $2x + 3y = 7$ and $4x - y = 5$.

Design a problem about a monthly mobile phone plan. The plan must involve a fixed monthly charge and a per-minute rate for extra talk time. The problem should require a user to formulate and solve a system of linear equations based on two different monthly bills. Finally, evaluate which of two given plans is more economical for a specific usage.

Problem Statement: A mobile phone company offers a plan with a fixed monthly charge and an additional charge for each extra minute of talk time. In April, a customer used 150 extra minutes and the total bill was ₹450. In May, the customer used 220 extra minutes and the total bill was ₹555. (a) Formulate a pair of linear equations to find the fixed monthly charge and the charge per extra minute. (b) Solve the system to find these charges. (c) The company offers a second plan with a fixed charge of ₹300 and a rate of ₹1.2 per extra minute. Evaluate which of the two plans is more economical for a usage of 300 extra minutes per month and justify your answer.

Summarize the three possible graphical representations for a pair of linear equations in two variables. For each case, describe the corresponding algebraic condition based on the ratios of their coefficients ($\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, $\frac{c_1}{c_2}$) and state the nature of their solutions (unique, infinitely many, or no solution).

Solve for $x$ and $y$: $\frac{x}{2} + \frac{y}{3} = 4$ and $\frac{x}{3} + \frac{y}{2} = \frac{19}{6}$.

Identify whether the pair of linear equations $4x - y = 5$ and $8x - 2y = 9$ is consistent or inconsistent. Explain how you know without solving them.

For what value of $k$ does the pair of equations $x - 2y = 3$ and $3x + ky = 1$ have a unique solution?

Solve the following pair of linear equations using the substitution method: $3x + 2y = 11$ and $2x + 3y = 4$.

The present age of a father is three years more than three times the age of his son. Three years hence, the father's age will be ten years more than twice the age of the son. Calculate their present ages.

A fraction becomes $\frac{1}{3}$ when 1 is subtracted from the numerator. It becomes $\frac{1}{4}$ when 8 is added to its denominator. Calculate the fraction.

Critique the following statement: "For a pair of linear equations to be consistent, the lines representing them must intersect at a single point."

A student claims that the system $y = 3x + 5$ and $y = -2x + 5$ represents coincident lines because the constant term is the same in both equations. Evaluate this claim.

Identify if the pair of equations $x + y = 5$ and $2x + 2y = 10$ represents intersecting, parallel, or coincident lines.

Describe the relationship between the coefficients of a pair of linear equations ($a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$) when their graphs are coincident lines.

Summarize the steps of the substitution method for solving a pair of linear equations.

For the pair of equations $2x + 3y - 7 = 0$ and $4x + 6y - 14 = 0$, compare the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$ and state the type of solution they have.

Explain the elimination method step-by-step. Describe what happens in this method if the pair of equations is (a) dependent (infinitely many solutions) and (b) inconsistent (no solution).

A taxi service in a city has a fixed charge along with a charge for the distance covered. For a journey of 12 km, the charge paid is ₹125. For a journey of 20 km, the charge paid is ₹197. Calculate the fixed charge and the charge per km. How much will a person have to pay for travelling a distance of 30 km?

The sum of a two-digit number and the number obtained by reversing its digits is 99. If the digits differ by 3, calculate the number. (Two possible answers exist, find both).

Design a pair of linear equations whose graphical representation results in intersecting lines, with the point of intersection being the origin $(0, 0)$. Justify your design.

A student solved the system $x - 2y = 8$ and $3x - 6y = 16$ and concluded there are infinitely many solutions because $3(x - 2y) = 3(8)$ gives $3x - 6y = 24$, which is "close" to the second equation. Critique the student's reasoning and provide the correct algebraic interpretation.

Design a word problem involving the cost of two items, for which the corresponding pair of linear equations would be $2x + 5y = 120$ and $4x + 10y = 200$. Justify why the scenario described in your problem results in no possible solution.

Propose a system of two linear equations whose solution is the point $(-2, 5)$. Then, create a third linear equation that also passes through this point, such that it forms a dependent (coincident) system with one of the original equations. Justify your creations.

The area of a rectangle gets reduced by 9 square units if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Calculate the dimensions of the rectangle.

For the pair of equations $2x + 3y = 7$ and $(k-1)x + (k+2)y = 3k$, justify the value of $k$ for which the pair will have infinitely many solutions.

Calculate the values of $a$ and $b$ for which the following pair of linear equations has infinitely many solutions: $2x + 3y = 7$ and $(a-b)x + (a+b)y = 3a+b-2$.

Solve the following pair of linear equations using the elimination method: $4x + 3y = 25$ and $3x - 4y = -4$.

Justify the choice of the elimination method over the substitution method for solving the system: $7x + 11y = 29$ and $11x + 7y = 25$.

Given the equation $3x - 2y = 6$, write another linear equation in two variables such that the pair represents parallel lines. Explain your reasoning.

The sum of a two-digit number and the number formed by interchanging its digits is 110. If 10 is subtracted from the original number, the new number is 4 more than 5 times the sum of the digits of the original number. Formulate a pair of linear equations for this problem and find the number. Justify that your solution is unique.

Explain what it means algebraically when the substitution method results in a false statement like $0 = 5$.

Critique the graphical method as a universal tool for solving linear equations. Formulate a specific pair of linear equations for which this method would be impractical and justify your choice.

Consider the general form of a pair of linear equations: $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$. Explain the geometric meaning of each of the following conditions: (i) $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ (ii) $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ (iii) $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ Also, relate each condition to the number of solutions the system has.

Evaluate the statement: "A system of linear equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ will always have a unique solution, unless the lines are parallel." Critique this statement and then prove that if $\frac{a_1}{a_2} = \frac{b_1}{b_2}$, the system cannot have a unique solution.

Formulate and solve a system of linear equations for the following problem: A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water.