NCERT Class 9 Maths Solutions: Linear Equations in Two Variables
Question:
Which one of the following options is true, and why?
y = 3x + 5 has
A. a unique solution
B. only two solutions
C. infinitely many solutions
Concept in a Minute:
Linear equations in two variables have infinitely many solutions. A linear equation in two variables is an equation of the form Ax + By = C, where A, B, and C are constants and A and B are not both zero. The solutions to such an equation are pairs of values (x, y) that satisfy the equation. Graphically, a linear equation in two variables represents a straight line, and every point on that line is a solution.
Explanation:
The given equation is y = 3x + 5. This is a linear equation in two variables, x and y. For any real value we choose for x, we can find a corresponding real value for y that satisfies the equation. For example:
If x = 0, then y = 3(0) + 5 = 5. So, (0, 5) is a solution.
If x = 1, then y = 3(1) + 5 = 8. So, (1, 8) is a solution.
If x = -1, then y = 3(-1) + 5 = 2. So, (-1, 2) is a solution.
Since we can choose any real number for x, and for each such choice, there is a unique corresponding value of y, there are infinitely many possible pairs of (x, y) that satisfy the equation. Therefore, the equation y = 3x + 5 has infinitely many solutions.
The final answer is $\boxed{C}$.
Question:
Write four solutions for the following equation:
2x + y = 7
Concept in a Minute:
Linear equations in two variables have infinitely many solutions. To find a solution, we can choose a value for one variable and then solve for the other.
Explanation:
The given equation is 2x + y = 7. This is a linear equation in two variables, x and y. We need to find four pairs of (x, y) that satisfy this equation. We can do this by picking a value for either x or y and then calculating the corresponding value for the other variable.
Let’s find four solutions:
Solution 1:
Choose x = 1.
Substitute x = 1 into the equation:
2(1) + y = 7
2 + y = 7
Subtract 2 from both sides:
y = 7 – 2
y = 5
So, one solution is (x, y) = (1, 5).
Solution 2:
Choose x = 2.
Substitute x = 2 into the equation:
2(2) + y = 7
4 + y = 7
Subtract 4 from both sides:
y = 7 – 4
y = 3
So, another solution is (x, y) = (2, 3).
Solution 3:
Choose y = 1.
Substitute y = 1 into the equation:
2x + 1 = 7
Subtract 1 from both sides:
2x = 7 – 1
2x = 6
Divide by 2:
x = 6 / 2
x = 3
So, another solution is (x, y) = (3, 1).
Solution 4:
Choose y = -1.
Substitute y = -1 into the equation:
2x + (-1) = 7
2x – 1 = 7
Add 1 to both sides:
2x = 7 + 1
2x = 8
Divide by 2:
x = 8 / 2
x = 4
So, another solution is (x, y) = (4, -1).
Therefore, four solutions for the equation 2x + y = 7 are (1, 5), (2, 3), (3, 1), and (4, -1).
Question:
Write four solutions for the following equation:
πx + y = 9
Concept in a Minute:
A linear equation in two variables, like “ax + by = c”, has infinitely many solutions. A solution is a pair of values for x and y that makes the equation true. To find solutions, we can choose a value for one variable and then solve for the other.
Explanation:
The given equation is πx + y = 9. This is a linear equation with two variables, x and y. To find four solutions, we can pick any four values for x and then calculate the corresponding values for y. Alternatively, we can pick any four values for y and calculate the corresponding values for x.
Let’s choose four different values for x and find the corresponding y values:
Solution 1:
Let x = 0.
Substitute x = 0 into the equation:
π(0) + y = 9
0 + y = 9
y = 9
So, one solution is (0, 9).
Solution 2:
Let x = 1.
Substitute x = 1 into the equation:
π(1) + y = 9
π + y = 9
y = 9 – π
So, another solution is (1, 9 – π).
Solution 3:
Let x = 2.
Substitute x = 2 into the equation:
π(2) + y = 9
2π + y = 9
y = 9 – 2π
So, a third solution is (2, 9 – 2π).
Solution 4:
Let x = -1.
Substitute x = -1 into the equation:
π(-1) + y = 9
-π + y = 9
y = 9 + π
So, a fourth solution is (-1, 9 + π).
Therefore, four solutions for the equation πx + y = 9 are (0, 9), (1, 9 – π), (2, 9 – 2π), and (-1, 9 + π).
Question:
Write four solutions for the following equation:
x = 4y
Concept in a Minute:
Linear equations in two variables have infinitely many solutions. A solution is a pair of values (x, y) that satisfies the equation. To find solutions, we can choose a value for one variable and then calculate the corresponding value for the other variable using the given equation.
Explanation:
The given equation is x = 4y.
This is a linear equation in two variables, x and y.
To find solutions, we can pick any value for ‘y’ and then substitute it into the equation to find the corresponding value of ‘x’.
Let’s find four solutions:
Solution 1:
Choose y = 1.
Substitute y = 1 into the equation x = 4y.
x = 4 * 1
x = 4
So, one solution is (x, y) = (4, 1).
Solution 2:
Choose y = 2.
Substitute y = 2 into the equation x = 4y.
x = 4 * 2
x = 8
So, another solution is (x, y) = (8, 2).
Solution 3:
Choose y = 0.
Substitute y = 0 into the equation x = 4y.
x = 4 * 0
x = 0
So, another solution is (x, y) = (0, 0).
Solution 4:
Choose y = -1.
Substitute y = -1 into the equation x = 4y.
x = 4 * (-1)
x = -4
So, another solution is (x, y) = (-4, -1).
Therefore, four solutions for the equation x = 4y are (4, 1), (8, 2), (0, 0), and (-4, -1).
Question:
Express the following linear equation in the form ax + by + c = 0 and indicate the values of a, b and c in the case:
3x + 2 = 0
Concept in a Minute:
A linear equation in two variables is an equation that can be written in the form ax + by + c = 0, where a, b, and c are real numbers, and at least one of a or b is non-zero. This form is called the standard form of a linear equation. To express a given equation in this form, we need to rearrange its terms so that all terms are on one side of the equation, set equal to zero.
Explanation:
The given equation is 3x + 2 = 0.
This equation is a linear equation in one variable, x. However, the question asks to express it in the form ax + by + c = 0, which is the standard form for a linear equation in *two* variables, x and y.
To fit the given equation into this form, we can consider the coefficient of y to be zero.
We can rewrite the equation 3x + 2 = 0 as:
3x + 0y + 2 = 0
Now, this equation is in the form ax + by + c = 0.
By comparing 3x + 0y + 2 = 0 with ax + by + c = 0, we can identify the values of a, b, and c.
Here,
a is the coefficient of x, which is 3.
b is the coefficient of y, which is 0.
c is the constant term, which is 2.
Therefore, the values are:
a = 3
b = 0
c = 2
Question:
Express the following linear equation in the form ax + by + c = 0 and indicate the values of a, b and c in the case:
2x = –5y
Concept in a Minute:
The standard form of a linear equation in two variables is ax + by + c = 0, where a, b, and c are constants and a and b are not both zero. To express an equation in this form, all terms must be moved to one side of the equation, resulting in zero on the other side.
Explanation:
The given linear equation is 2x = –5y.
To express this equation in the form ax + by + c = 0, we need to move all the terms to one side.
Add 5y to both sides of the equation:
2x + 5y = –5y + 5y
2x + 5y = 0
Now, we compare this equation with the standard form ax + by + c = 0.
We can see that:
The coefficient of x is a. In our equation, the coefficient of x is 2. So, a = 2.
The coefficient of y is b. In our equation, the coefficient of y is 5. So, b = 5.
The constant term is c. In our equation, there is no constant term, which means the constant term is 0. So, c = 0.
Therefore, the linear equation 2x = –5y expressed in the form ax + by + c = 0 is 2x + 5y + 0 = 0.
The values of a, b, and c are a = 2, b = 5, and c = 0.
Question:
Express the following linear equation in the form ax + by + c = 0 and indicate the values of a, b and c in the case:
y – 2 = 0
Concept in a Minute:
A linear equation in two variables is an equation that can be written in the standard form ax + by + c = 0, where a, b, and c are real numbers and at least one of a or b is not zero. The question asks to rewrite a given linear equation in this standard form and identify the coefficients a, b, and c.
Explanation:
The given linear equation is y – 2 = 0.
We need to express this in the form ax + by + c = 0.
Observe that the term with x is missing in the given equation. This means the coefficient of x is 0. So, we can write 0x for the x term.
The term with y is present as +y, which means the coefficient of y is 1. So, we can write 1y.
The constant term is -2. So, we can write c = -2.
Combining these, we get 0x + 1y + (-2) = 0, which simplifies to 0x + y – 2 = 0.
Comparing this with the standard form ax + by + c = 0, we can identify the values of a, b, and c.
Here, a = 0, b = 1, and c = -2.
Final Answer: The final answer is $\boxed{0x + 1y – 2 = 0; a=0, b=1, c=-2}$
Question:
The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement.
(Take the cost of a notebook to be ₹ x and that of a pen to be ₹ y).
Concept in a Minute:
Understanding the relationship between two quantities and expressing it as an algebraic equation. Forming a linear equation in two variables by identifying the variables and the given condition.
Explanation:
The problem states that the cost of a notebook is twice the cost of a pen.
We are given that the cost of a notebook is represented by ₹ x.
We are also given that the cost of a pen is represented by ₹ y.
The statement “the cost of a notebook is twice the cost of a pen” can be translated into a mathematical equation.
“is” translates to the equals sign (=).
“twice the cost of a pen” means 2 times the cost of the pen.
So, the cost of a notebook = 2 × (cost of a pen).
Substituting the given variables:
x = 2 × y
x = 2y
This is a linear equation in two variables, x and y. We can also write it in the standard form Ax + By + C = 0 by rearranging:
x – 2y = 0
The linear equation in two variables representing the statement is x = 2y or x – 2y = 0.
Question:
Find the value of k, if x = 2, y = 1 is a solution of the equation 2x + 3y = k.
Concept in a Minute:
Substitution: If a point (x, y) is a solution to an equation, it means that when you substitute the x and y values into the equation, the equation holds true.
Algebraic Manipulation: Once values are substituted, the equation can be solved for the unknown variable using basic algebraic operations.
Explanation:
The problem states that x = 2 and y = 1 is a solution to the equation 2x + 3y = k.
This means that if we substitute the given values of x and y into the equation, the equation will be satisfied.
Step 1: Write down the given equation.
2x + 3y = k
Step 2: Substitute the given values of x and y into the equation.
Given x = 2 and y = 1.
Substitute these values into the equation:
2(2) + 3(1) = k
Step 3: Perform the multiplication.
2 * 2 = 4
3 * 1 = 3
Step 4: Substitute the results back into the equation.
4 + 3 = k
Step 5: Perform the addition.
7 = k
Step 6: State the value of k.
Therefore, the value of k is 7.
The final answer is 7.
Question:
Express the following linear equation in the form ax + by + c = 0 and indicate the values of a, b and c in the case:
x = 3y
Concept in a Minute:
The standard form of a linear equation in two variables is ax + by + c = 0, where a, b, and c are constants, and a and b are not both zero. To express a given linear equation in this form, we need to rearrange its terms so that all terms are on one side of the equation and the other side is zero.
Explanation:
The given linear equation is x = 3y.
To express this equation in the form ax + by + c = 0, we need to move all terms to one side of the equation and set the other side to zero.
Subtract 3y from both sides of the equation:
x – 3y = 3y – 3y
x – 3y = 0
Now, compare this equation with the standard form ax + by + c = 0.
We can see that:
The coefficient of x is 1, so a = 1.
The coefficient of y is -3, so b = -3.
There is no constant term, which means the constant term is 0, so c = 0.
Therefore, the linear equation x = 3y expressed in the form ax + by + c = 0 is x – 3y + 0 = 0.
The values of a, b, and c are:
a = 1
b = -3
c = 0
Question:
Express the following linear equation in the form ax + by + c = 0 and indicate the values of a, b and c in the case:
–2x + 3y = 6
Concept in a Minute:
A linear equation in two variables is an equation that can be written in the standard form ax + by + c = 0, where a, b, and c are constants, and at least one of a or b is non-zero. The goal is to rearrange the given equation into this specific format.
Explanation:
The given linear equation is -2x + 3y = 6.
To express it in the form ax + by + c = 0, we need to move all the terms to one side of the equation, making the other side equal to zero.
We can achieve this by subtracting 6 from both sides of the equation:
-2x + 3y – 6 = 6 – 6
-2x + 3y – 6 = 0
Now, the equation is in the form ax + by + c = 0.
By comparing -2x + 3y – 6 = 0 with ax + by + c = 0, we can identify the values of a, b, and c.
The coefficient of x is a, so a = -2.
The coefficient of y is b, so b = 3.
The constant term is c, so c = -6.
Therefore, the linear equation -2x + 3y = 6 can be expressed as -2x + 3y – 6 = 0, with a = -2, b = 3, and c = -6.
Question:
Express the following linear equation in the form ax + by + c = 0 and indicate the values of a, b and c in the case:
5 = 2x
Concept in a Minute:
A linear equation in two variables is an equation that can be written in the standard form ax + by + c = 0, where a, b, and c are constants and a and b are not both zero. The goal is to rearrange the given equation to match this form and then identify the coefficients a, b, and c.
Explanation:
The given equation is 5 = 2x.
To express this in the form ax + by + c = 0, we need to move all terms to one side of the equation.
We can rewrite the equation as:
2x – 5 = 0
Now, compare this to the standard form ax + by + c = 0.
We can see that the term with x is 2x, so the coefficient of x, which is ‘a’, is 2.
There is no term with y in the equation, which means the coefficient of y, ‘b’, is 0.
The constant term is -5, so ‘c’ is -5.
Therefore, the linear equation in the form ax + by + c = 0 is 2x + 0y – 5 = 0.
The values of a, b, and c are:
a = 2
b = 0
c = -5
Question:
Check whether the following is the solution of the equation x – 2y = 4 or not:
(4, 0)
Concept in a Minute:
To check if a given ordered pair (x, y) is a solution to a linear equation, substitute the values of x and y from the ordered pair into the equation. If the equation holds true (i.e., both sides of the equation are equal), then the ordered pair is a solution. Otherwise, it is not.
Explanation:
The given equation is x – 2y = 4.
The given ordered pair is (4, 0).
In the ordered pair (4, 0), the value of x is 4 and the value of y is 0.
Substitute these values into the equation:
4 – 2(0) = 4
Now, perform the calculation on the left side of the equation:
4 – 0 = 4
4 = 4
Since the left side of the equation is equal to the right side of the equation, the ordered pair (4, 0) is a solution to the equation x – 2y = 4.
Question:
Check whether the following is the solution of the equation x – 2y = 4 or not:
(2, 0)
Concept in a Minute:
To check if a given ordered pair (x, y) is a solution to a linear equation in two variables, substitute the values of x and y from the ordered pair into the equation. If the equation holds true (i.e., the left-hand side equals the right-hand side), then the ordered pair is a solution. Otherwise, it is not.
Explanation:
The given equation is x – 2y = 4.
The given ordered pair is (2, 0).
Here, x = 2 and y = 0.
Substitute these values into the equation:
Left-hand side (LHS) = x – 2y
LHS = 2 – 2(0)
LHS = 2 – 0
LHS = 2
Right-hand side (RHS) = 4
Now, compare the LHS and RHS:
LHS = 2
RHS = 4
Since LHS (2) is not equal to RHS (4), the ordered pair (2, 0) is not a solution to the equation x – 2y = 4.
Question:
Check whether the following is the solution of the equation x – 2y = 4 or not:
(0, 2)
Concept in a Minute:
To check if a given ordered pair (x, y) is a solution to a linear equation, substitute the values of x and y from the ordered pair into the equation. If the left-hand side (LHS) of the equation equals the right-hand side (RHS), then the ordered pair is a solution. Otherwise, it is not.
Explanation:
The given equation is x – 2y = 4.
The given ordered pair is (0, 2). This means x = 0 and y = 2.
Substitute these values into the equation:
LHS = x – 2y
LHS = (0) – 2(2)
LHS = 0 – 4
LHS = -4
The RHS of the equation is 4.
Now, compare the LHS and RHS:
LHS = -4
RHS = 4
Since LHS ≠ RHS (-4 ≠ 4), the ordered pair (0, 2) is not a solution to the equation x – 2y = 4.
Answer: No
Question:
Check whether the following is the solution of the equation x – 2y = 4 or not:
(1, 1)
Concept in a Minute:
To check if a given point (x, y) is a solution to a linear equation, substitute the x and y values from the point into the equation. If the equation holds true (i.e., the left side equals the right side), then the point is a solution. Otherwise, it is not.
Explanation:
The given equation is x – 2y = 4.
The given point is (1, 1). This means x = 1 and y = 1.
Substitute these values into the equation:
1 – 2(1) = 4
1 – 2 = 4
-1 = 4
Since -1 is not equal to 4, the point (1, 1) is not a solution to the equation x – 2y = 4.
Next Chapter: Lines and Angles
Refer Linear Equations in Two Variables Notes
Practice Linear Equations in Two Variables Extra Questions
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