Pair of Linear Equations in Two Variables — Important Questions
25 questions
With answersCBSE format
SUMMARY: This chapter focuses on solving pairs of linear equations in two variables using various methods. KEY TOPICS: graphical method, substitution method, elimination method, cross-multiplication method, consistency of equations, algebraic methods, applications of linear equations, word problems, system of equations, solutions of linear equations
Which of the following methods can be used to solve the pair of linear equations 2x + 3y = 6 and 4x - y = 5?
AGraphical method
BSubstitution method
CElimination method
DAll of the above
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Correct answer: Option 4 — All of the above
Q21 Mark
If the equations 3x + 4y = 12 and 6x + 8y = 24 are given, what can be inferred about their consistency?
AThey are inconsistent.
BThey are dependent.
CThey are independent.
DThey have no solution.
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Correct answer: Option 2 — They are dependent.
Q31 Mark
Using the substitution method, if x = 2y + 3 is substituted into the equation 4x - 5y = 7, what is the resulting equation in terms of y?
A8y + 12 - 5y = 7
B8y - 12 - 5y = 7
C4(2y + 3) - 5y = 7
D4(2y + 3) + 5y = 7
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Correct answer: Option 3 — 4(2y + 3) - 5y = 7
Q41 Mark
In the graphical method, what does the point of intersection of two lines represent?
AA solution to the equations
BThe slope of the lines
CThe x-intercept of the lines
DThe y-intercept of the lines
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Correct answer: Option 1 — A solution to the equations
Q51 Mark
If a pair of linear equations has no solution, what type of lines do they represent in a graph?
ACoincident lines
BParallel lines
CIntersecting lines
DPerpendicular lines
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Correct answer: Option 2 — Parallel lines
Short Answer Questions5 questions
Q63 Marks
Solve the pair of linear equations using the substitution method: 2x + 3y = 6 and x - y = 1.
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From the second equation, x = y + 1. Substituting into the first gives 2(y + 1) + 3y = 6. Simplifying, we find y = 0 and substituting back gives x = 1. Thus, the solution is (1, 0).
Q73 Marks
Explain the graphical method for solving a pair of linear equations in two variables.
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The graphical method involves plotting the equations on a graph. The point where the two lines intersect represents the solution to the equations. If the lines coincide, there are infinitely many solutions; if they are parallel, there is no solution.
Q83 Marks
What is the elimination method for solving linear equations? Provide an example.
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The elimination method involves adding or subtracting equations to eliminate one variable. For example, for the equations 3x + 4y = 10 and 2x + 4y = 8, subtracting the second from the first eliminates y, allowing us to solve for x.
Q93 Marks
Determine the consistency of the following equations: 4x + 2y = 8 and 2x + y = 4. Are they consistent or inconsistent?
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To check for consistency, we can see if the second equation is a multiple of the first. Dividing the first equation by 2 gives 2x + y = 4, which is the same as the second equation. Therefore, the equations are consistent and have infinitely many solutions.
Q103 Marks
A word problem states that the sum of two numbers is 30 and their difference is 10. Formulate the equations and solve them using the cross-multiplication method.
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Let the two numbers be x and y. The equations are x + y = 30 and x - y = 10. Using the cross-multiplication method, we can express x and y in terms of each other and find that x = 20 and y = 10. Thus, the numbers are 20 and 10.
Long Answer Questions5 questions
Q116 Marks
Solve the following pair of linear equations using the substitution method: 2x + 3y = 12 and x - y = 1. Show all steps involved in your solution.
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To solve the equations 2x + 3y = 12 and x - y = 1 using the substitution method, first solve the second equation for x: x = y + 1. Substitute this expression for x into the first equation: 2(y + 1) + 3y = 12. Simplifying gives 2y + 2 + 3y = 12, or 5y + 2 = 12. Solving for y yields 5y = 10, so y = 2. Substituting y back into x = y + 1 gives x = 3. Therefore, the solution is (3, 2).
Q126 Marks
Using the graphical method, solve the following pair of linear equations: 3x - 2y = 6 and 6x + 4y = 24. Explain how you would graph these equations and find the point of intersection.
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To solve the equations 3x - 2y = 6 and 6x + 4y = 24 using the graphical method, first rewrite both equations in slope-intercept form. For the first equation, rearranging gives y = (3/2)x - 3. For the second equation, simplifying gives y = -3/2x + 12. Next, plot both lines on a graph. The first line has a y-intercept of -3 and a slope of 3/2, while the second line has a y-intercept of 12 and a slope of -3/2. The point of intersection can be found by graphing the lines accurately, which shows they intersect at the point (6, 6).
Q136 Marks
Demonstrate the elimination method to solve the following pair of linear equations: 4x + 5y = 20 and 2x - 3y = -6. Include all necessary calculations.
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To solve the equations 4x + 5y = 20 and 2x - 3y = -6 using the elimination method, first multiply the second equation by 2 to align the coefficients of x: 4x - 6y = -12. Now we have the system: 4x + 5y = 20 and 4x - 6y = -12. Subtract the second equation from the first: (4x + 5y) - (4x - 6y) = 20 - (-12), which simplifies to 11y = 32. Solving for y gives y = 32/11. Substitute y back into one of the original equations to find x: 4x + 5(32/11) = 20, leading to x = (20 - 160/11)/4. Thus, the solution is (x, y) = (5/11, 32/11).
Q146 Marks
Explain the concept of consistency of equations in the context of linear equations. Provide an example of a consistent and an inconsistent pair of linear equations.
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The consistency of equations refers to whether a pair of linear equations has at least one solution. A consistent pair has one unique solution or infinitely many solutions, while an inconsistent pair has no solutions. For example, the equations 2x + 3y = 6 and 4x + 6y = 12 are consistent because they represent the same line, thus having infinitely many solutions. In contrast, the equations x + y = 2 and x + y = 5 are inconsistent, as they represent parallel lines that never intersect, resulting in no solutions.
Q156 Marks
A word problem states that the sum of two numbers is 30, and their difference is 10. Formulate the pair of linear equations and solve them using any method of your choice.
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Let the two numbers be x and y. The problem states that the sum of the numbers is 30, which gives us the equation x + y = 30. The difference of the numbers is 10, leading to the equation x - y = 10. We can solve this pair of equations using the elimination method. Adding both equations gives 2x = 40, thus x = 20. Substituting x back into the first equation gives 20 + y = 30, leading to y = 10. Therefore, the two numbers are 20 and 10.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): The graphical method can be used to find the solution of a pair of linear equations in two variables.
Reason (R): The point of intersection of the two lines represents the solution of the equations.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q171 Mark
Assertion (A): If a pair of linear equations has no solution, they are called consistent equations.
Reason (R): Inconsistent equations represent parallel lines which do not intersect.
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Correct answer: Option 4 —
A is false, but R is true.
Q181 Mark
Assertion (A): The substitution method is effective when one of the equations can be easily solved for one variable.
Reason (R): This method involves substituting the value of one variable into the other equation.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q191 Mark
Assertion (A): The elimination method can be used to solve a pair of equations even if the coefficients of one variable are equal.
Reason (R): In such cases, the equations can still be manipulated to eliminate that variable.
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Correct answer: Option 2 —
Both A and R are true, but R is not the correct explanation of A.
Q201 Mark
Assertion (A): Cross-multiplication method is applicable only when both equations are in standard form.
Reason (R): This method requires the equations to be expressed in the form ax + by = c.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: The graphical method can be used to find the solution of a pair of linear equations in two variables.
Statement 2: The elimination method involves substituting one variable in terms of another.
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Correct answer: Option 2 —
Only Statement 1 is true.
Q221 Mark
Statement 1: A pair of linear equations is said to be consistent if it has at least one solution.
Statement 2: The cross-multiplication method can only be applied when the equations are in standard form.
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Correct answer: Option 1 —
Both statements are true.
Q231 Mark
Statement 1: The substitution method is always the best method to solve linear equations regardless of the situation.
Statement 2: If two lines intersect at a point, the system of equations is consistent and has a unique solution.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q241 Mark
Statement 1: The elimination method can be used to solve equations that are not in standard form.
Statement 2: A pair of linear equations can have infinitely many solutions if they are dependent.
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Correct answer: Option 1 —
Both statements are true.
Q251 Mark
Statement 1: Graphical representation of linear equations can help in visualizing the solutions.
Statement 2: The substitution method is ineffective for solving equations with no common variables.
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Correct answer: Option 2 —
Only Statement 1 is true.
