Linear Functions – You know about straight lines. The equation of a straight line is y = mx+j. The mathematical expression representing the straight line is called a linear function. So y = mx+j represents the linear function where y is the dependent variable. x is the independent variable that we manipulate to get different values of y. m is the coefficient of the independent variable or the slope-intercept. It determines the rate of change of y. j is the constant term.

The graph of a linear function is a straight line. To plot a graph of linear function you have to follow the steps given below

- You need to find two points that satisfy the equation y = mx+j.
- Now plot the graph for these points.
- Connect the points to get a straight line.

To get hands-on experience of graph plotting attend math classes conducted by Cuemath. For more details, you can visit the Cuemath website.

## Applications of Linear Equations

Linear equations make use of one or more variables. Here a variable is dependent on the other. Almost in any situation, you can use linear equations. Ex: For-profit prediction, fuel consumption rate, etc. Some of its applications include

- Solving geometric problems by using two variables.
- Solving budget-related problems
- Solving Distance-Rate-Time related problems
- While predicting the market in business and economics

Let’s practice some examples to understand linear equations. To know this concept better you can also log in to Cuemath website.

## Solved Examples on Linear Equations:

Before solving the problems, we must know the equation to find the slope of a line. Usually, Slope is defined as the ratio of vertical change to horizontal change. The formulas of the slope are m = tanθ and m = (y_{2} – y_{1})/(x_{2} – x_{1}) Where (x_{1}, y_{1}), (x_{2}, y_{2}) are the points on the line and θ is the angle subtended by the line with the horizontal.

- Find the slope of a graph for the following function.

f(2) = -2 and f(6) = 6

Solution: The ordered pair of the given function is (2, -2) and (6, 6).

By using slope formula we have, m = where x_{1} = 2, x_{2} = 6, y_{1} = -2 and y_{2} = 6.

m = = = 2

Hence the slope of the graph is 2.

- The Sum of two Numbers is 120. If the first number is twice the second number, find the numbers.

Solution: Let the two numbers be p and q. Then P + Q = 120. But P is twice Q, that is P = 2Q.

Now we can write 2Q + Q = 120 3Q = 120

So Q = 40 and P = 2 40 = 80.

Verifying P+Q = 120

80+40 = 120. Hence the two numbers are 80 and 40 respectively.

- The current Age of Sam is one-fourth the age of his father. After 5 years, he becomes one-third of his father’s age. Then, calculate their present ages.

Solution: Let the current age of Sam be x and his fathers be y. Then x = ¼ y 4x = y.

After 5 years, the age of the son = x + 5 years and the age of father = y + 5 years.

But after 5 years the son’s age (x+5) = ⅓ of the father’s age (y+5) (x+5) = ⅓ (y+5)

Or 3x+15 = y+5 now by substituting 4x = y we get,

(3x+15) = (4x+5) 4x – 3x = 15 – 5

x = 10 and y = 4x = 40. After 5 years 3r+15 = y+5.

3 10 + 15 = 40 + 5

45 = 45. LHS = RHS

Hence the current age of Sam is 10 years and that of the father is 40 years.