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 = (y2 – y1)/(x2 – x1) Where (x1, y1), (x2, y2) 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 x1 = 2, x2 = 6, y1 = -2 and y2 = 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.
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