The linear equations questions and answers will assist students to understand the concepts better. Linear Equation is a topic that is covered in basically every class. The NCERT guidelines will be followed for preparing the questions. Linear Equations are used in mathematics as well as in everyday life. So, the basics of this concept must be grasped. For students of all levels, the problems here will include both the basics and more challenging problems. As a result, students will be able to use it to solve problems involving linear equations. Learn more about Linear Equations by clicking here.
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Definition: A linear equation is defined as an algebraic equation in which each term has an exponent of 1 and when graphed, the result is always a straight line. In other words, a linear equation is an equation with a maximum degree of 1. |
Here, we’ll go through a variety of linear equations problems with solutions, based on various concepts.
Linear Equations Questions with Solutions
1. Write the statement as an equation: A number increased by 8 equals 15.
Solution:
Given statement: A number increased by 8 equals 15.
Let the number be “x”.
So, x is increased by 8 means x + 8.
Hence, x increased by 8 equals 15 means x + 8 = 15, which is the equation for the given statement.
2. Write the statement for the given equation: 2x = 18.
Solution:
Given equation: 2x = 18.
The statement for the given equation is “Twice the number x equals 18”.
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Linear Equations in One Variable: An equation with only one variable is known as a linear equation in one variable. It’s written as Ax + B = 0, with A and B being any two integers and x being an unknown variable only with one solution. Also, read: Linear Equation in One Variable. |
3. Solve the equation: x + 3 = -2
Solution:
Given equation: x + 3 = -2.
Now, keep the variables on one side and constants on the other side. Hence, the equation becomes,
x = -2 -3
x = -5
Hence, the value of x is -5.
4. Verify that x = 4 is the root of the equation 3x/2 = 6.
Solution:
To verify whether the given root is the solution of the given equation, substitute x = 4 in the equation 3x/2 = 6.
⇒ (3(4))/2 = 6
⇒ (12/2) = 6
⇒ 6 = 6
Hence, x = 4 is the root of the equation 3x/2 = 6.
5. If 5 is added to twice a number, the result is 29. Determine the number.
Solution:
The equation for the given statement is 5+2x = 29.
To find the number “x”, we have to solve the equation.
⇒ 2x = 29 – 5
⇒ 2x = 24
⇒ x = 24/2
⇒ x = 12
Hence, the required number is 12.
6. If x = 2, then 2x – 5 = 7. Check whether the statement is true or false.
Solution:
Given equation: 2x – 5 = 7
If x = 2,
= 2(2) – 5
= 4 – 5 = -1
Hence, the given statement is false.
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Linear Equations in Two Variables: The standard form of linear equations in two variables is Ax + By + C = 0, in which A, B, and C are constants and x and y are the two variables and each variable with a degree of 1. When two linear equations are evaluated at the same time, they are referred to as simultaneous linear equations. Also, read: Linear Equations in Two Variables. |
7. The sum of two consecutive numbers is 11. Find the numbers.
Solution:
Let the number be x.
Hence, the two consecutive numbers are x and x+1.
According to the given statement, the equation becomes
⇒ x + x + 1 = 11
⇒ 2x + 1 = 11
⇒ 2x = 10
⇒ x = 10/2 = 5
If x = 5, then x + 1 = 5 + 1 = 6
Hence, the two numbers are 5 and 6.
8. Express the equation x = 3y in the form of ax+by+c = 0 and find the values of a, b and c.
Solution:
Given equation: x = 3y
We know that the standard form of linear equation in two variables is ax+by+c = 0 …(1)
Now, rearranging the given equation, we get
⇒ x – 3y = 0
This can be written as
⇒ 1(x) + (-3)y + (0)c = 0 …(2)
On comparing equation (1) and (2), we get
⇒ a = 1, b = -3 and c = 0.
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Also, read: |
9. Find three solutions for the equation 2x + y = 7.
Solution:
To find the solutions for the equation 2x + y = 7, substitute different values for x.
When x = 0,
⇒ 2(0) + y = 7
⇒ y = 7
Therefore, the solution is (0, 7).
When x = 1,
⇒ 2(1) + y = 7
⇒ y = 7 – 2
⇒ y = 5
Hence, the solution is (1, 5).
When x = 2,
⇒ 2(2) + y = 7
⇒ 4 + y = 7
⇒ y = 3
Hence, the solution is (2, 3).
Therefore, the three solutions are (0, 7), (1, 5) and (2, 3).
10. Solve the following equations using the substitution method:
3x + 4y = 10 and 2x – 2y = 2
Solution:
3x + 4y = 10 …(1)
2x – 2y = 2 …(2)
Equation (2) can be written as:
2(x – y) = 2
x – y = 1
x = 1+y …(3)
Now, substitute (3) in (1), we get
3 (1+y) + 4y = 10
3 + 3y + 4y = 10
7y = 10 – 3
7y = 7
Hence, y = 1.
Now, substitute y = 1 in (3), we get
x = 1 + 1
x = 2.
Hence, x = 2 and y = 1 are the solutions of the given equations.
Practice Questions
- Write the statement as an equation: Twice a number subtracted from 19 is 11.
- The sum of the two numbers is 30 and their ratio is 2: 3. Find the numbers.
- If the point (3, 4) lies on the graph of equation 3y = ax + 7, determine the value of a.
- Solve the equations using the elimination method: (x/2)+(2y/3) = -1 and x – (y/3) = 3.
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