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Questions and Answers
What does the equation $3x + 4y = 20$ represent in Akhila's scenario?
What does the equation $3x + 4y = 20$ represent in Akhila's scenario?
- Total number of Hoopla games played
- Total amount of money earned from games
- Total number of rides Akhila took
- Total cost of rides and games (correct)
A pair of linear equations that has a solution is referred to as an inconsistent pair.
A pair of linear equations that has a solution is referred to as an inconsistent pair.
False (B)
If Akhila had 2 rides, how many times did she play Hoopla?
If Akhila had 2 rides, how many times did she play Hoopla?
1
A pair of linear equations which has no solution is called an _________ pair of linear equations.
A pair of linear equations which has no solution is called an _________ pair of linear equations.
Match the terms with their definitions:
Match the terms with their definitions:
If Akhila spent all her money, which equation describes her total spending?
If Akhila spent all her money, which equation describes her total spending?
Two equations that represent the same line are considered independent.
Two equations that represent the same line are considered independent.
If the number of rides Akhila took is represented by x, then the number of times she played Hoopla is represented by _______.
If the number of rides Akhila took is represented by x, then the number of times she played Hoopla is represented by _______.
What is the solution to the system of equations 2x + 3y = 11 and 2x – 4y = –24?
What is the solution to the system of equations 2x + 3y = 11 and 2x – 4y = –24?
The pair of linear equations for ‘The difference between two numbers is 26 and one number is three times the other’ can be represented as x - y = 26 and x = 3y.
The pair of linear equations for ‘The difference between two numbers is 26 and one number is three times the other’ can be represented as x - y = 26 and x = 3y.
What is the value of 'm' if y = mx + 3 and the point (4, 11) lies on this line?
What is the value of 'm' if y = mx + 3 and the point (4, 11) lies on this line?
The larger of two supplementary angles exceeds the smaller by ____ degrees.
The larger of two supplementary angles exceeds the smaller by ____ degrees.
Match the following real-world situations with their corresponding equations:
Match the following real-world situations with their corresponding equations:
What do the taxi charges for 15 km amount to if for 10 km the charge is ₹105?
What do the taxi charges for 15 km amount to if for 10 km the charge is ₹105?
If the equation 0.2x + 0.3y = 1.3 is solved for x and y, it will always yield integer solutions.
If the equation 0.2x + 0.3y = 1.3 is solved for x and y, it will always yield integer solutions.
What is the fixed charge for the taxi if the charge per km is assessed from the earlier data?
What is the fixed charge for the taxi if the charge per km is assessed from the earlier data?
What does it mean if the elimination method results in a false statement?
What does it mean if the elimination method results in a false statement?
If an elimination results in a true statement involving no variable, the equations have a unique solution.
If an elimination results in a true statement involving no variable, the equations have a unique solution.
What is the first step in using the elimination method?
What is the first step in using the elimination method?
In the elimination method, if you obtain the statement _____ involving no variable, the original pair of equations has infinitely many solutions.
In the elimination method, if you obtain the statement _____ involving no variable, the original pair of equations has infinitely many solutions.
Match the following outcomes with their meanings in the context of equation pairs:
Match the following outcomes with their meanings in the context of equation pairs:
When reversing the digits of a two-digit number, what happens to the ten's and unit's digits?
When reversing the digits of a two-digit number, what happens to the ten's and unit's digits?
What is the general form of a two-digit number where 'x' is the ten's digit and 'y' is the unit's digit?
What is the general form of a two-digit number where 'x' is the ten's digit and 'y' is the unit's digit?
In Example 9, multiplying the first equation by 2 results in valid linear equations to work with.
In Example 9, multiplying the first equation by 2 results in valid linear equations to work with.
What is the charge for each extra day for the book Saritha and Susy borrowed?
What is the charge for each extra day for the book Saritha and Susy borrowed?
If two lines are parallel, the equations they represent are consistent.
If two lines are parallel, the equations they represent are consistent.
What method can be used to determine the unique solution of two linear equations?
What method can be used to determine the unique solution of two linear equations?
When the equations of two lines coincide, they have __________ solutions.
When the equations of two lines coincide, they have __________ solutions.
Match the following situations of linear equations with their outcomes:
Match the following situations of linear equations with their outcomes:
What is the value of x when solving the equations x + y = 6 and x - y = 2?
What is the value of x when solving the equations x + y = 6 and x - y = 2?
The numbers 42 and 24 both sum to 66 and have digits that differ by 2.
The numbers 42 and 24 both sum to 66 and have digits that differ by 2.
What are the two numbers that satisfy the equations derived from the initial problem?
What are the two numbers that satisfy the equations derived from the initial problem?
The sum of the digits of a two-digit number is _____ when the number is 9.
The sum of the digits of a two-digit number is _____ when the number is 9.
Match the following methods with their corresponding equations:
Match the following methods with their corresponding equations:
If a fraction reduces to 1 by adding 1 to the numerator and subtracting 1 from the denominator, what is the reduced equation?
If a fraction reduces to 1 by adding 1 to the numerator and subtracting 1 from the denominator, what is the reduced equation?
Five years ago, Nuri was twice as old as Sonu.
Five years ago, Nuri was twice as old as Sonu.
Meena received a total of _____ notes from the cashier.
Meena received a total of _____ notes from the cashier.
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Study Notes
Pair of Linear Equations in Two Variables
- A pair of linear equations in two variables can be represented by two lines.
- If the lines intersect at a point, the point represents the unique solution of the two equations.
- This is called a consistent pair of linear equations.
- If the lines coincide, there are infinitely many solutions. Each point on the line is a solution. This is called a dependent (consistent) pair of linear equations.
- If the lines are parallel, the pair of equations has no solution. This is called an inconsistent pair of linear equations.
Solving Pair of Linear Equations
- Graphical Method:
- Requires plotting the lines represented by the equations.
- The point of intersection is the solution.
- Substitution Method:
- Solve one equation for one variable.
- Substitute the expression for the variable into the other equation.
- Solve the resulting equation for the remaining variable.
- Substitute the found value back into one of the original equations to find the other variable.
- Elimination Method:
- Multiply the equations by constants so that the coefficients of one variable are the same or opposite.
- Add or subtract the equations to eliminate one variable.
- Solve the resulting equation in one variable.
- Substitute the value of that variable into one of the original equations to find the other variable.
Special Cases:
a₁b₁
- Case 1:
- ≠ a₂b₁
- This indicates a consistent pair of linear equations. a₁b₁c₁
- Case 2:
- = a₂b₂ ≠ c₂
- This indicates an inconsistent pair of linear equations. a₁b₁c₁
- Case 3:
- = a₂b₂ = c₂
- This indicates a dependent and consistent pair of linear equations.
Real-World Applications of Linear Equations
- Linear equations can model various real-world problems like determining the number of rides and games at a fair, finding the age of individuals, or calculating the cost of items.
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