Linear Equations in Two Variables — Important Questions
25 questions
With answersCBSE format
SUMMARY: This chapter introduces the concept of linear equations in two variables and explores their graphical representation and solutions. KEY TOPICS: linear equation, two variables, Cartesian plane, graph of a linear equation, solution of a linear equation, intercepts, slope, parallel lines, coincident lines, algebraic methods of solving equations
What is the general form of a linear equation in two variables?
AAx + By + C = 0
BAx^2 + By^2 = C
CA + B = C
Dy = mx + b
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Correct answer: Option 1 — Ax + By + C = 0
Q21 Mark
Which of the following represents the slope of the line in the equation y = mx + b?
Ab
Bm
Cx
Dy
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Correct answer: Option 2 — m
Q31 Mark
If the linear equations 2x + 3y = 6 and 4x + 6y = 12 are plotted on a graph, what type of lines will they represent?
AParallel lines
BCoincident lines
CIntersecting lines
DPerpendicular lines
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Correct answer: Option 2 — Coincident lines
Q41 Mark
What are the x-intercept and y-intercept of the linear equation 3x + 4y = 12?
A(4, 0) and (0, 3)
B(0, 4) and (3, 0)
C(0, 3) and (4, 0)
D(3, 0) and (0, 4)
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Correct answer: Option 3 — (0, 3) and (4, 0)
Q51 Mark
Which of the following methods can be used to solve a system of linear equations?
AGraphical method
BSubstitution method
CElimination method
DAll of the above
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Correct answer: Option 4 — All of the above
Short Answer Questions5 questions
Q63 Marks
Define a linear equation in two variables and provide an example.
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A linear equation in two variables is an equation that can be expressed in the form ax + by + c = 0, where a, b, and c are constants, and x and y are the variables. An example is 2x + 3y - 6 = 0.
Q73 Marks
What is the graphical representation of a linear equation in two variables?
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The graphical representation of a linear equation in two variables is a straight line on the Cartesian plane. Each point on the line represents a solution to the equation.
Q83 Marks
Explain the concept of intercepts in the context of linear equations.
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Intercepts are the points where the line intersects the axes. The x-intercept is the point where the line crosses the x-axis (y=0), and the y-intercept is where it crosses the y-axis (x=0).
Q93 Marks
How do you determine the slope of a line represented by a linear equation in two variables?
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The slope of a line represented by a linear equation in the form y = mx + b is given by the coefficient m of x. It indicates the steepness and direction of the line.
Q103 Marks
What distinguishes parallel lines from coincident lines in the context of linear equations?
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Parallel lines have the same slope but different y-intercepts, meaning they never intersect. Coincident lines have the same slope and y-intercept, meaning they lie on top of each other and have infinitely many solutions.
Long Answer Questions5 questions
Q116 Marks
Explain the concept of a linear equation in two variables. Provide an example and describe how to represent it graphically on a Cartesian plane.
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A linear equation in two variables is an equation that can be expressed in the form ax + by + c = 0, where a, b, and c are constants, and x and y are the variables. For example, the equation 2x + 3y - 6 = 0 is a linear equation in two variables. To represent this graphically on a Cartesian plane, we can find the intercepts by setting x and y to zero respectively. The graph will be a straight line, and every point on this line represents a solution to the equation.
Q126 Marks
Describe the method of finding the slope of a linear equation. How does the slope affect the graph of the equation?
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The slope of a linear equation is a measure of its steepness and is calculated as the ratio of the change in y to the change in x (rise over run). For a linear equation in the form y = mx + b, 'm' represents the slope. A positive slope indicates that the line rises as it moves from left to right, while a negative slope indicates that it falls. The slope affects the graph by determining the angle at which the line inclines or declines, influencing the relationship between the variables represented by the equation.
Q136 Marks
What are intercepts in the context of a linear equation? Calculate the x-intercept and y-intercept for the equation 4x - 2y = 8.
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Intercepts are points where the graph of a linear equation intersects the axes of the Cartesian plane. The x-intercept is found by setting y to zero, while the y-intercept is found by setting x to zero. For the equation 4x - 2y = 8, to find the x-intercept, set y = 0: 4x = 8, which gives x = 2. For the y-intercept, set x = 0: -2y = 8, leading to y = -4. Thus, the x-intercept is (2, 0) and the y-intercept is (0, -4).
Q146 Marks
Differentiate between parallel lines and coincident lines in the context of linear equations. Provide examples of each.
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Parallel lines are lines in a Cartesian plane that never intersect and have the same slope but different y-intercepts. For example, the equations y = 2x + 1 and y = 2x - 3 represent parallel lines. Coincident lines, on the other hand, are lines that lie on top of each other, meaning they have the same slope and y-intercept. An example of coincident lines would be the equations 3x + 4y = 12 and 6x + 8y = 24, as the second equation is a multiple of the first, indicating they represent the same line.
Q156 Marks
Explain the algebraic methods of solving linear equations in two variables. Illustrate with an example using the substitution method.
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Algebraic methods for solving linear equations in two variables include substitution, elimination, and cross-multiplication. The substitution method involves solving one equation for one variable and substituting that expression into the other equation. For example, consider the equations x + y = 10 and x - y = 2. From the first equation, we can express y as y = 10 - x. Substituting this into the second equation gives x - (10 - x) = 2, which simplifies to 2x - 10 = 2, leading to x = 6. Substituting x back into y = 10 - x gives y = 4. Thus, the solution is (6, 4).
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): A linear equation in two variables can be represented graphically as a straight line on the Cartesian plane.
Reason (R): The solution of a linear equation in two variables is a set of points that satisfy the equation.
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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): The slope of a line is the same for parallel lines.
Reason (R): Parallel lines never intersect each other.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q181 Mark
Assertion (A): The x-intercept of a linear equation is the point where the line crosses the y-axis.
Reason (R): The x-intercept occurs when y is equal to zero.
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Correct answer: Option 3 —
A is true, but R is false.
Q191 Mark
Assertion (A): Coincident lines have different slopes.
Reason (R): Coincident lines represent the same linear equation.
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Correct answer: Option 4 —
A is false, but R is true.
Q201 Mark
Assertion (A): The graphical representation of a linear equation can help in finding its solutions.
Reason (R): Each point on the line represents a solution to the equation.
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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: A linear equation in two variables can be represented as ax + by + c = 0 where a, b, and c are constants.
Statement 2: The graph of a linear equation in two variables is always a parabola.
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Correct answer: Option 2 —
Only Statement 1 is true.
Q221 Mark
Statement 1: The slope of a line is defined as the change in y divided by the change in x between two points on the line.
Statement 2: Parallel lines have different slopes.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q231 Mark
Statement 1: The x-intercept of a linear equation is the point where the graph intersects the x-axis.
Statement 2: The solution of a linear equation in two variables is a unique point on the Cartesian plane.
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Correct answer: Option 1 —
Both statements are true.
Q241 Mark
Statement 1: Two lines are said to be coincident if they have the same slope and y-intercept.
Statement 2: If two lines are parallel, they will intersect at exactly one point.
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Correct answer: Option 1 —
Both statements are true.
Q251 Mark
Statement 1: To find the intercepts of a linear equation, we can set y=0 to find the x-intercept and x=0 to find the y-intercept.
Statement 2: The graphical representation of a linear equation in two variables can be a curve.
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Correct answer: Option 1 —
Both statements are true.
