Linear Equation
The equation given to draw a straight line is called a Linear Equation. By drawing a straight line it means that if we assume different values of x and y, then it will form a table with us and then we will find the coordinates for a line in that table. A linear equation is any equation that can be written in the form “ax+b=0” where “a” and “b” are real numbers and “x” is a variable. This form is sometimes called the standard form of a linear equation. Students can check all details about Linear Equation Formula, Important Question Solutions & Examples and Tricks from this page.
Linear Equation Facts
- If a=b then a+c = b+c for any c. All this is saying is that we can add a number, c, to both sides of the equation and not change the equation.
- If a=b then a−c = b−c for any c. As with the last property we can subtract a number, c, from both sides of an equation.
- If a=b then ac=bc for any c. Like addition and subtraction, we can multiply both sides of an equation by a number, c, without changing the equation.
- If a=b thenfor any non-zero c. We can divide both sides of an equation by a non-zero number, c, without changing the equation.
Linear Equation Formulas
Standard Formula Of Linear Equation
The Linear equation formula is given by
y=mx+b
Where,
m determines the slope of that line,
b determines the point at which the line crosses
Formulas of Linear equations in one variable
- A Linear Equation in one variable is defined as ax + b = 0
- Where, a and b are constant, a ≠ 0, and x is an unknown variable
- The solution of the equation ax + b = 0 is x = . We can also say that is the root of the linear equation ax + b = 0.
Formulas of Linear equations in two variable
- A Linear Equation in two variables is defined as ax + by + c = 0
- Where a, b, and c are constants and also, both a and b ≠ 0
Formulas for Linear equations in three variable
- A Linear Equation in three variables is defined as ax + by + cz = d
- Where a, b, c, and d are constants and also, a, b and c ≠ 0
Key Points To Remember For Linear Equation
Suppose, there are two linear equations:
Now,
(A) If , then there will be one solution, and the graphs will have intersecting lines.
(B) If , then there will be numerous solutions, and the graphs will have coincident lines.
(C) If , then there will be no solution, and the graphs will have parallel lines.
Standard Form of a Linear Equation
Consider a, b, c, a1 , a2 , b1 , b2 and d are real numbers and x, y, z are variables.
Terms | Definition | Standard Form | Example |
Linear Equations in One Variable | An equation has only one variable. | ax + b = 0 | 2x – 3 = 0 |
Linear Equation in two Variables | An equation has only two variables. | ax + by + c = 0 | 3x + 7y + 4 = 0 |
Linear Equation in Three Variables | An equation has three variables. | ax + by + cz + d = 0 | x + 7y + 4z -1 = 0 |
Linear Equations with Fractions | Equation containing fractional terms. | a/x + b/y = c | 2/x – 3/y = 5 and 3/x = 1/y + 1 |
Slope Intercept Form | Equation in the form: y = mx + c. | y = mx + c, where ‘m’ = slope and ‘c’ = y-intercept | y = 3x + 10 |
Point Slope Form | Equation passing through a point (x1, y1). | y − y1 = m(x − x1) where m = slope | y – 3 = 10(x – 5); Line passing through point (3, 5) |
Pair of linear equations in two variables | Set of two or more linear equations containing same number of variables | Pair of 2 equations: a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 | 3x + 4y = 4 and 5x + 7y = 3 |
Tricks For Linear Equation
- Linear Equations Questions can be easily solved by eliminating the wrong options. It means put the given values in equation and check which one is satisfying the equation.
- Standard form of linear equations is y= mx+b
- There are 2 types of questions asked in exams explain below.
Type 1: Linear Equations Tips and Tricks and Shortcuts. To Find the value of x or y
Question 1. If 3a+6 = 4a−2, then find the value of a?
Options:
- 3
- 8
- 6
- 7
Solution: We can use the trick of eliminating the option
Option 1, put a = 3
3 * 3 + 6 = 15
4 * 3 -2 = 10
This means option 1 is incorrect.
Now, check for option 2, put a = 8
3 * 8 + 6 = 30
4 * 8 – 2 = 30
This means option 2 satisfies the equation. Therefore, it is the correct option.
Correct option: B
Type 2: Tips And Tricks And Shortcuts For Linear Questions Word Problems
Question 2. The cost of 5 blankets and 6 bedsheets is Rs.1500. The cost of 6 blankets and 5 bedsheets is Rs.1300. Find out the total cost of one blanket and one bedsheet.
Options:
- Rs. 255
- Rs. 250
- Rs. 81.81
- Rs. 254.545
Solution: Let the cost of blankets be x and the cost of bedsheets be y.
According to the question:
5x+ 6y= 1500…(1)
6x+ 5y=1300…(2)
Multiply Eq 1 by 5 and Eq 2 by 6,
we get.
25x+30y = 7500…(3)
36x+30y = 7800…(4)
Subtract equation (3) from equation (4)
11x = 300
x =
5 ×
6y =
6y =
6y =
y =
Total cost = x+y
=>
=>
= 254.545
Correct option: D
How to Solve Linear Equations
- If the equation includes any fractions then use the least common denominator to clear the fractions.
- Simplify both sides of the equation by clearing out any parenthesis and then further combining like terms.
- Now get all terms with the variable in them on one side of the equations and all constants on the other side.
- If the coefficient of the variable is not unit value then make the coefficient a one.
- We can verify the answer if we wish to check. We verify the answer by substituting the results from the previous steps into the original equation.
Linear Equation Important Questions with Solutions
Solve the following equations and check your answer:
Ques1. 4x – 7 (2 – x) = 3x + 2
Solution: Value of x = 2
Explanation: 4x – 7 (2 – x) = 3x + 2 4x – 14 + 7x = 3x + 2 11x−14=3x+2 8x=16 x=2 |
Ques2. 2(w+3)−10 = 6(32−3w)
Solution: Value of w = 49/5
Explanation: 2(w+3)−10 = 6(32−3w) 2w+6−10 = 192−18w 2w−4 = 192−18w 20w=196 W=49/5 |
Ques3. x−7y = −11
5x+2y = −18
Solution: x = −4,y = 1
Explanation: x−7y = −11 ⇒x = 7y−11 5x+2y = −18 5(7y−11)+2y = −18 5(7y−11)+2y = −18 35y−55+2y = −18 37y=37 =>y = 1 Put value of y in equation: X = 7(1)−11 = −4 Therefore, x = −4,y = 1 |
Ques4. 7x−8y = −12
−4x+2y=3
Solution: x = 0, y = 32
Explanation: −4x+2y=32 y=4x+3 ⇒y=2x+3/2 7x−8y= −12 7x−8(2x+32)= −12 7x−8(2x+32)= −12 7x−16x−12= −12 −9x =0 =>x=0 Put Value of x in equation: Y = 2(0) + 3/2 = 3/2 |
Ques5.
Solution:
Explanation: |
Ques6.
Solution: Value of t = -1
Explanation: Multiply for both sides with (t−5) (t+5) |
Ques7.
Solution:
Explanation: Multiply for both sides with (y−1) |
Ques8. 6x−5y=8
−12x+2y=0
Solution: Value of x= −1/3, y= −2
Explanation: |
Ques9. 3x+9y=−6
−4x−12y=8
Solution:
Explanation: |
Ques10.
Solution:
Explanation: |
Linear Equations Word Problems with Solution
Ques1. The sum of three consecutive multiples of 4 is 444. Find these multiples.
Answer: Three consecutive multiples of 4 are 144, 148, 152
Explanation: If x is a multiple of 4, the next multiple is x + 4, next to this is x + 8. Their sum = 444 According to the question, x + (x + 4) + (x + 8) = 444 ⇒ x + x + 4 + x + 8 = 444 ⇒ x + x + x + 4 + 8 = 444 ⇒ 3x + 12 = 444 ⇒ 3x = 444 – 12 ⇒ x = 432/3 ⇒ x = 144 Therefore, x + 4 = 144 + 4 = 148 Therefore, x + 8 – 144 + 8 – 152 Therefore, the three consecutive multiples of 4 are 144, 148, 152. |
Ques2. The denominator of a rational number is greater than its numerator by 3. If the numerator is increased by 7 and the denominator is decreased by 1, the new number becomes 3/2. Find the original number.
Answer: The original number is 8/11
Explanation: Let the numerator of a rational number = x Then the denominator of a rational number = x + 3 When numerator is increased by 7, then new numerator = x + 7 When denominator is decreased by 1, then new denominator = x + 3 – 1 The new number formed = 3/2 According to the question, (x + 7)/(x + 3 – 1) = 3/2 ⇒ (x + 7)/(x + 2) = 3/2 ⇒ 2(x + 7) = 3(x + 2) ⇒ 2x + 14 = 3x + 6 ⇒ 3x – 2x = 14 – 6 ⇒ x = 8 The original number i.e., x/(x + 3) = 8/(8 + 3) = 8/11 |
Ques3. The sum of the digits of a two digit number is 7. If the number formed by reversing the digits is less than the original number by 27, find the original number.
Answer: The original number is 52
Explanation: Let the units digit of the original number be x. Then the tens digit of the original number be 7 – x Then the number formed = 10(7 – x) + x × 1 = 70 – 10x + x = 70 – 9x On reversing the digits, the number formed = 10 × x + (7 – x) × 1 = 10x + 7 – x = 9x + 7 According to the question, New number = original number – 27 ⇒ 9x + 7 = 70 – 9x – 27 ⇒ 9x + 7 = 43 – 9x ⇒ 9x + 9x = 43 – 7 ⇒ 18x = 36 ⇒ x = 36/18 ⇒ x = 2 Therefore, 7 – x = 7 – 2 = 5 The original number is 52 |
Ques4. A motorboat goes downstream in river and covers a distance between two coastal towns in 5 hours. It covers this distance upstream in 6 hours. If the speed of the stream is 3 km/hr, find the speed of the boat in still water.
Answer: Required speed of the boat is 33 km/hr.
Explanation: Let the speed of the boat in still water = x km/hr. Speed of the boat downstream = (x + 3) km/hr. Time taken to cover the distance = 5 hrs Therefore, distance covered in 5 hrs = (x + 3) × 5 (D = Speed × Time) Speed of the boat upstream = (x – 3) km/hr Time taken to cover the distance = 6 hrs. Therefore, distance covered in 6 hrs = 6(x – 3) Therefore, the distance between two coastal towns is fixed, i.e., same. According to the question, 5(x + 3) = 6(x – 3) ⇒ 5x + 15 = 6x – 18 ⇒ 5x – 6x = -18 – 15 ⇒ -x = -33 ⇒ x = 33 Required speed of the boat is 33 km/hr. |
Ques5. Divide 28 into two parts in such a way that 6/5 of one part is equal to 2/3 of the other.
Answer: The Two parts are 10 & 18.
Explanation: Let one part be x. Then other part = 28 – x It is given 6/5 of one part = 2/3 of the other. ⇒ 6/5x = 2/3(28 – x) ⇒ 3x/5 = 1/3(28 – x) ⇒ 9x = 5(28 – x) ⇒ 9x = 140 – 5x ⇒ 9x + 5x = 140 ⇒ 14x = 140 ⇒ x = 140/14 ⇒ x = 10 Then the two parts are 10 and 28 – 10 = 18. |
Note:
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