Content Writer
A linear equation is a type of equation which consists of two expressions that are assigned to the same value. It depends on the number of variables and coefficients of the equations.
- A linear equations is an algebraic equation in which each component has a value 1.
- When plotted on a graph, the result is always a straight line.
- It can be expressed in terms of one, two, and three variables.
- The coefficients are called the parameters of the equation.
- Linear equations are the popular name for one-degree equations.
- The equations are solved by equating zero with a linear polynomial.
- Equation written on the left side are called left-hand side.
- The equation written on the right side is called right-hand side.
Read More: Pair of Linear equation in two variables formula
Key Terms: Linear Equations, Positive Slope, Negative Slope, Variables, Coefficients, Zero Slope, Linear Polynomial, One-degree Equations, Solution
Linear Equations in One Variable
[Click Here for Sample Questions]
Linear equations in one variable are equations that are written as ax+b = 0, where a and b are two integers and x is a variable.
ax + b = 0
- The numerals 'a' and 'b' are real.
- Neither 'a' nor 'b' is equal to zero.
- The solution for the given linear equation is x = -b/a.
Read More:
Solved Example of Linear Equations in One VariableExample: Consider a linear equation of the form 2x + 3 = 8, Calculate the value of x. Solution: 2x + 3 = 8 2x = 8 – 3 2x = 5 x = 5/2 |
Linear equation terms
Solving Linear Equations in One Variable
The steps for solving a one-variable equation are as follows:
- Remove any fractions using LCM.
- Make both sides of the equation simpler.
- Remove the variable from the equation.
- Double-check your response.
Read More: Horizontal and Vertical Lines
Linear Equations in Two Variables
[Click Here for Sample Questions]
Linear Equations in Two Variables is defined as a linear relationship between x and y, or two variables in which the value of one (typically y) relies on the value of the other (usually x). Because x is the independent variable and y is dependent on it, y is referred to as the dependent variable in this situation.
- There are two solutions to a linear equation with two variables.
- The graph of linear equation in two variable represent a straight line.
- The independent variable is normally plotted along the horizontal axis, regardless of whether it is labelled x or not.
- To put it another way, there is only one y value for every x value.
- Compute the value of the dependent variable, y, once the independent variable, x, has been assigned a value.
- After that, you may use a coordinate grid to plot the points specified by each (x,y) pair.
- The usual form of linear equations (general form of linear equations) in two variables.
Ax + By = C
- Where x and y are variables, while A, B, and C are integers.
Read More: Line Segment
Solved Example of Linear Equations in Two VariablesExample: The usual form of a two-variable linear equation is 3x + 4y = 8. Consider the following scenario: 30 = 5x + 3y The equation above involves two variables: x and y. This equation can be represented graphically by setting the variables to zero. When y = 0, the value of x is 30 = 5x + 3(0) x = 6 and when x = 0 the value of y is, 30 = 5 (0) + 3y y = 10 |
Solving Linear Equations in Two Variables
There are four ways to solve linear equations in two variables which are as follows:
- Elimination Method
- Graphical Method
- Cross Multiplication Method
- Substitution Method
Read More: Coincident Lines
Linear Equations in Three Variables
[Click Here for Sample Questions]
Linear Equations in Three Variables is defined as the type of equation which defines relationship between three variables x, y and z. It is a type of equation in which the value of the two variables (typically z and y) relies on the value of the other (usually x). It can be represented as:
ax+ by + cz + d = 0
- where x, y and z are variables, while a, b, and c are integers.
- where a ≠ 0, b ≠ 0, c ≠ 0, x, y, z are the variables.
Solved Example of Linear Equations in Two VariablesExample: The point x = 3, y = 0, and z = 1 is a solution of the following system of three linear equations in three variables 3x + 2y 5z = 14 2x 3y + 4z = 10 x + y + z = 4 That’s because we can substitute 3, 0, and 1 for x, y, and z respectively in the equations above and check that -3(3) + 2(0) - 5(1) = - 9 - 5 = -14 2(3) – 3(0) + 4(1) = 6 + 4 = 10 (3) + (0) + (1) = 4 |
Solving Linear Equations in Three Variables
There are four ways to solve linear equations in two variables which are as follows:
- Elimination Method
- Graphical Method
- Cross Multiplication Method
- Substitution Method
Read More: Perpendicular Line Formula
Formula for a Linear Equation
[Click Here for Sample Questions]
A line's equation can be expressed in such a way that the slope is clear and the line can be drawn without any calculations. Students can construct linear equations in slope-intercept form if they are familiar with solving a simple two-step linear equation. The slope intercept form of linear equations is given by,
y = mx + b
- The variables in the equation are x and y. When x is 0, the integers m and b represent the line's slope (m) and the value of y. (b).
- Because (0,y) is the point where the line crosses the y-axis, the value of y when x is 0 is called the y-intercept
Equation of Line
By plotting (0,b) and then using m to find another point, you may draw the line for an equation that matches this linear formula. If m is 1/2, for example, you can think of it as a difference of 1 in y coordinates for every difference of 2 in x coordinates (that is, (y2 – y1)/(x2 – x1) = 1/2).
- To go to another point, count +2 on the x-axis and then +1 on the y-axis: (2, b + 1).
- This line's equation is y + 3 = 2x. The equation is y = 2x – 3 in slope-intercept form.
- The slope m = 2 is seen in this form.
- According to the graph, the slope is 2 since every +2 change in y corresponds to a +1 change in x.
Read More: Distance between Two Points
Types of Slope
[Click Here for Sample Questions]
Types of Slope are divided into three categories, which are discussed in detail in the section below:
Zero Slope
The graph of the line is horizontal when y does not change while x changes. The slope of a horizontal line is zero.
Zero Slope
Read More: Properties of Parallel Lines
Positive slope
Positive slope is when a line slopes up from left to right. This implies that an increase in y is accompanied by an increase in x. The greater the rate of change in y in respect to the change in x, the steeper the slope.
- A slope of 6 is steeper than a slope of 1, and a slope of 1/6 is steeper still.
- A positive slope shows a positive correlation when the line reflects real-world data points.
- It is displayed on a coordinate plane, and the steeper the slope, the stronger the positive correlation.
Positive Slope
Example of Positive slopeConsider a linear equation in which the dependent variable d is the distance travelled in miles and the independent variable g is the number of gallons of gas utilised. You get low gas mileage if you drive a huge, old car.
|
Read More: Straight line
Negative slope
Negative slope is when a line slopes down from left to right. This implies that a decrease in y is accompanied by an increase in x. A negative slope implies a negative correlation when the line reflects real-world data points drawn on a coordinate plane, and the steeper the slope, the stronger the negative correlation.
Negative slope
Example of Negative slopeConsider a line that indicates the number of peppers left to plant after gardening for a certain amount of time. The rate at which the garden flat empties out is fairly high if the garden can contain 18 pepper plants.
|
Read More:
Things to Remember
- Linear Equations is an algebraic equation with the degree of variable is 1.
- The solution or root of a linear equation is the variable's value that makes it true.
- The result is unaltered when the same number is added, subtracted, multiplied, or divided.
- A linear equations with one or two variables always forms a straight-line graph.
- In the equation, no variable is raised to a power larger than one or used as the fraction denominator.
- All the points on a coordinated grid lie on the same line.
Read More: Equation of a line
Sample Questions
Ques. What is the definition of a linear equation? (2 marks)
Ans. The equations of degree one are known as linear equations. It's the formula for a straight line. ax+by+c =0 is the conventional form of a linear equation, where a and b are both 0.
Ques. What is the formula for one-variable linear equations? (2 marks)
Ans. A linear equation in one variable has the usual form Ax + B = 0. Here, A represents the x coefficient, x represents the variable, and B represents the constant term. To determine the final solution of this linear equation, separate the coefficient and the constant term.
Ques. How can you tell if an equation in one variable is linear? (2 marks)
Ans. If the graph of an equation is a straight line, it is said to be linear. When the largest power of x is 1, this will occur. If the equation produces a straight line graphically, it is a linear equation. Otherwise, if you get a circle, a parabola, or any other conic, it's a quadratic or nonlinear equation.
Ques. What is the purpose of a linear equation? (2 marks)
Ans. Linear equations can be used to describe a wide range of physical interactions and processes, and hence serve an important role in research. Linear equations are frequently used to calculate rates, such as the speed at which a projectile moves or the rate at which a chemical reaction occurs.
Ques. In two variables, what is a linear equation? (2 marks)
Ans. If an equation is written in the form ax+by+c=0, the coefficients of x and y, i.e. a and b, are not equal to zero; it is said to be a linear equation in two variables. Linear equations in two variables such as 10x+4y = 3 and -x+5y = 2 are examples.
Ques. Solve the equation x = 12(x +2) ? (3 marks)
Ans. 12(x + 2) = x
12x + 24 = x
On both sides of the equation, subtract 24.
x – 24 = 12x + 24 – 24
12x = x – 24
Simplify
-24 = 11x
Remove x from the equation:
x = -24/11
Ques. Solve: 4x−7(2−x)=3x+2? (2 marks)
Ans.
4x - 7(2 - x) = 3x + 2
4x - 14 + 7x = 3x + 2
11x - 14 = 3x + 2
8x = 16
x = 2
Hence, x=2
Ques. Solve: (3 marks)
Ans.
Ques. The sum of two numbers is 44. If one number is 20 more than the other, find the numbers by framing a linear equation? (3 marks)
Ans. Let the number be x, so the other number is x + 20.
- We know that the sum of both numbers is 44.
- Therefore, the linear equation can be framed as, x + x + 20 = 44.
- This results in, 2x + 20 = 44.
- Now, let us solve the equation by isolating the variable on one side and by bringing the constants on the other side.
- This means 2x = 44 - 20.
- By simplifying RHS, we get, 2x = 24, so the value of x is 12.
- This means, one number is 12 and the other number is 12 + 10 = 22.
Ques. Six times of a number is equal to 42. Find the linear equation that corresponds to the situation and find the unknown number? (3 marks)
Ans. Let the unknown number be x.
- Six times of this number is equal to 42.
- This gives the linear equation 6x = 42.
- So, this linear equation can be solved to find the value of x which is the unknown number.
- 6x = 42
- x = 42/6
- x = 7.
Ques. Solve the equation x = 11(x +3) ? (3 marks)
Ans. 11(x + 3) = x
11x + 33 = x
On both sides of the equation, subtract 33.
x – 33 = 11x + 33 – 33
11x = x – 33
Simplify
-33 = 10x
Remove x from the equation:
x = -33/10
Ques. Calculate the linear equation for x: 5x - 90 = 70? (2 marks)
Ans. The given equation is 5x - 90 = 70.
⇒ 5x = 70 + 90
⇒ 5x = 160
⇒ x = 32
Ques. In the last three months Mr.Sharma lost 5 and a half kg, gained 2 and one-fourth kg and then lost 3 and three fourth kg weight. If he now weighs 90 kg, then how much did Mr.Sharma weighs in beginning? (2 marks)
Ans. Let x be the Mr.Sharma’s weight in beginning. According to given in instructions,
x – 5.5 + 2.5 – 3.75 = 90
⇒ x = 90 + 5.5 + 3.75 – 2.25 = 90 + 7 = 97 kg.
Ques. The sum of two numbers is 80. If one number is 40 more than the other, find the numbers by framing a linear equation? (3 marks)
Ans. Let the number be x, so the other number is x + 40.
- We know that the sum of both numbers is 80.
- Therefore, the linear equation can be framed as, x + x + 40 = 80.
- This results in, 2x + 40 = 80.
- Now, let us solve the equation by isolating the variable on one side and by bringing the constants on the other side.
- This means 2x = 80 - 40.
- By simplifying RHS, we get, 2x = 40, so the value of x is 20.
- This means, one number is 20 and the other number is 20 + 10 = 30.
Ques. Eight times of a number is equal to 160. Find the linear equation that corresponds to the situation and find the unknown number? (3 marks)
Ans. Let the unknown number be x.
- Six times of this number is equal to 160.
- This gives the linear equation 8x = 160.
- So, this linear equation can be solved to find the value of x which is the unknown number.
- 8x = 160
- x = 160/8
- x = 20.
Check Out:
Comments