We can use linear equations in our day-to-day life, for example, when comparing rates, making predictions, budgeting, and more. An equation is said to be linear when all the variables have the highest power equal to one. The variables are dependent on one another. This type of equation is also called a one-degree equation. Read on to discover the linear equation standard form, formula, graph, and guidelines to solve a linear equation in one or two variables.
Here is what we will cover in the article:
- What is a linear equation?
- Linear equation standard form
- Linear equation graph
- Linear equations in one variable
- Linear equations in two variables
- How do solve linear equations?
What is a Linear Equation?
An equation with the highest degree of 1 is linear. It has no variable with an exponent of more than 1. A linear equation always forms a straight line on a graph, and thus, it gets its name ‘linear equation’.
In Mathematics, we have linear equations in one variable and two variables. The following examples will help you learn how to differentiate linear equations from nonlinear ones.
|Equations||Linear or Nonlinear|
|y = 7x – 8||Linear|
|y = x3 – 6||Non-Linear as the power of the variable x is 3.|
|√y + x = 5||Non-Linear because the power of y is ½.|
|y + 5x – 3 = 0||Linear|
Linear Equation Formula
The linear equation formula expresses a linear equation. There are different ways a linear equation can be expressed. For example, we have the standard form, the point-slope form, or the slope-intercept form.
Linear Equations in Standard Form
The standard form of linear equations in one variable can be given as follows:
Ax + B = 0
The terms in the above equation denote:
- A and B are real numbers
- x is the single variable
The standard form of linear equations in two variables can be given as follows:
Ax + By = C
The terms in the above equation denote:
- A B and C are real numbers.
- A is the coefficient of x
- B is the coefficient of y
- C is constant
- x and y are the variables
Linear Equation Graph
On graphing a linear equation in one variable x, we get a vertical line parallel to the y-axis. If we graph a linear equation in two variables, x, and y, it forms a straight line. To graph, a linear equation, follow the steps given below.
- Step 1: Note down your linear equation and convert it into the form of y = mx + b.
- Step 2: When we have our equation in this form, we can replace the value of x for various numbers. We will get the resulting value of y, and we can create the coordinates.
- Step 3: Next, we will list the coordinates in tabular form.
- Step 4: Finally, we will plot the coordinates on a graph. Join the points to get a straight line. This line will represent our linear equation.
How to Solve Linear Equations?
The main goal of solving an equation is to balance the two sides. An equation resembles a weighing balance as we have to ensure that both sides weigh equally. So, if we add a number on one side, we must add it to the other side as well.
Similarly, if we divide or multiply a number on the left-hand side, we will do the same for the right-hand side. To solve a linear equation, we will bring the variables to one side and keep the constant on the other side. Then we will find the value of the unknown variable.
Tips on Solving Linear Equations:
- The solution or root of the linear equation is the value of the variable. It makes a linear equation true.
- This solution remains unaffected if the same number is multiplied, added, subtracted, or divided for both sides of the equation.
- On graphing a linear equation in one or two variables, we get a straight line.
Solution of Linear Equations in One Variable
We need to create a balance on both sides of the linear equation to solve it. The equality sign symbolizes that the expressions are equal on the two sides. The following sample linear equation will help you understand the steps of solving the linear equation in one variable.
Example 1: Solve (2x – 4)/2 = 3(x – 1)
Step 1: We will clear the fraction
x – 2 = 3(x – 1)
Step 2: We will simplify the two sides of the equations by opening the brackets and multiplying the number by the inner terms of the bracket.
x – 2 = 3x – 3
Step 3: Next, we will isolate x
3-2 = 3x – x
1 = 2x
½ = x
0.5 = x
Solution of Linear Equations in Two Variables
There are several methods for solving a linear equation in 2 variables. Some commonly used methods are:
- Method of substitution
- Cross multiplication method
- Method of elimination
We use the substitution method when we have two linear equations with two unknown values. The following steps will help you solve the linear equations.
- Step 1: First, simplify the given equation. Expand the parenthesis.
- Step 2: Now, we will solve one of the equations to obtain the value of either x or y.
- Step 3: Substitute the value of x in terms of y in the other equation or the value of y in terms of x.
- Step 4: Solve the new equation following the basic arithmetic operations rules (BODMAS/ DMAS) to find the value of a variable.
- Step 5: Finally, use the value obtained and find the value of the second variable.
Example 2: Calculate the value of x and y from the equations 2x+3y = 13 and x-2y = -4
2x + 3y = 13 —————— (i)
x-2y = -4 ———————– (ii)
Solving eq (ii)
x = 2y -4
Substituting the value of x in eq ( i)
2 ( 2y -4) + 3y = 13
4y -8 + 3y =13
7y = 13+8
Substituting the value of y in eq (ii)
x= 2 (3) -4
Cross Multiplication Method
Cross multiplication is one of the simplest methods, and it is applicable only when we are given a pair of linear equations in two variables. We multiply the numerator of one fraction to the denominator of the other. The denominator of the first term is multiplied by the numerator of another term. The following equation is for solving linear equations in two variables using the cross multiplication method.
The following steps will help you solve a linear equation:
- Step 1: First, we will multiply the given equations with non-zero constants. This process makes the coefficients of any one of the variables numerically equal.
- Step 2: Next, we will add or subtract one equation from the other. This step will eliminate one variable. Now, we will get an equation in one variable.
- Step 3: We can now solve the equation in one variable and get its value.
- Step 4: Once we have the value of a variable, we can substitute this value in any one of the equations to calculate the other variable.
Example 3: Solve the following equations for x and y
2x+3y=6 —————— (i)
We will add the two equations as follows
2x + 3y -2x +5y =6 +10
Since the coefficients of x are equal and opposite in sign, they will be eliminated.
8y = 16
Now, we will substitute the value of y in eq (i)
2x + 3(2) =6
2x + 6 = 6
Example 4: If the difference in the measures of the given two complementary angles is 22°. Find the measure of the two angles.
Solution: Let the angle be x. The complement of x = 90 – x
Given their difference = 22°
Therefore, (90 – x) – x = 22°
⇒ 90 – 2x = 22
⇒ -2x = 22 – 90
⇒ -2x = -68
⇒ x = 68/2
⇒ x = 34
The complementary angle will be 90 -34 = 56
Answer: The two complementary angles are 56 and 34.
Practice the following problems on linear equations to ace these questions in your examinations.
Question 1: Solve the following linear equations using the substitution method.
- 4x-3y=20 and 16x-6y=80
- 2x-5y=10 and 3x+8y=15
Question 2: Solve linear equations given below using the elimination method.
- 3x + y = 6 and 2x + 7y = 10.
- 4x + 2y = 5 and 4x + 6y =15
Question 3: The sum of two numbers is 55. Suppose one number exceeds the other by 8. Find the two numbers.
Question 4: The length of a rectangle is thrice its breadth. If the perimeter of the rectangle is 32 meters, find the length and breadth of the rectangle.
Question 5: James is five years younger than Lily. Four years later, Lily will be twice as old as James. What is their present age of James?
Question 6: The cost of three tables and two chairs is $605. If the table costs $50 more than the chair, what are the costs of the table and the chair?
Frequently Asked Questions
1. How to properly round numbers?
Ans. Rounding a number is a pretty common task, and it’s one that people often get wrong. Here’s how to do it right:
If the number ending in .5 is greater than or equal to 5, round up.
If the number ending in .5 is less than 5, but greater than or equal to zero, round down.
If the number ending in .5 is less than zero, round up.
2. When to round up numbers?
Ans. When you’re rounding up numbers, you have to make sure that you’re rounding them up appropriately. If the number is less than or equal to 5, round it down. If the number is greater than 5 but less than or equal to 10, round it up. If the number is greater than 10, round it up.
3. What are the rules in rounding off numbers?
Ans. Rounding off numbers is a fairly simple process. Here are the rules:
-If the number ends in 5, round up.
-If the number ends in 0, round up.
-If there are any remaining digits, round down.
4. What is the procedure for rounding off numbers?
Ans. Rounding off numbers is a simple process that you can use to make your calculations easier to understand. It’s the process of taking a number and removing some portion of it so that the result is closer to an integer. For example, say you have the number 45.5. To round it off, you would take half of 45 (22) and then add it back onto the number 45.5—this gives you 45.75, which is closer to an integer than 45.5 was before rounding occurred.
5. Should 5 be rounded up or down?
Ans. 5 should be rounded up. This is because if you round 5 down, you’ll lose one of the zeros that make it so special. You don’t want that.