Algebra – Linear Equations Applications

Introduction

Linear equations have numerous applications in mathematics. An equation may be defined as an algebraic expression equated to constant or other expressions. An equation with the highest exponential power of variables equal to one is called a linear equation. Mathematical expressions can be used to solve word problems. Mathematical knowledge is usually applied through word problems coded in the form of expressions. In this tutorial, we will understand linear equations, algebra of linear equations, solving linear equations, one equation, one variable, two equations, two variables, and some solved examples.

Linear Equations

An equation consisting of one or more variables having power equal to 1. A linear equation is also known as a one-degree equation.

Algebra of Linear Equations

A linear equation can be expressed in different forms, such as

• The standard form.

• The slope-intercept form.

• The point-slope form.

A linear equation is written in the algebraic form as $\mathrm{Ax\:+\:c\:=\:0\:or\:Ax\:+\:By\:+\:c\:=\:0}$. Where x and y are variables, A and B are the coefficients of variables x and y, respectively, and c is constant.

Solving Linear Equations

An equation has two sides: left-hand side (LHS) and right-hand side (RHS). When a number or a variable is added or subtracted from both sides of an equation, the equation remains true. Similarly, if we multiply or divide the LHS and RHS of the equation with the same variable or number, the equation remains true. A linear equation can be solved to find the value of variables.

For example − $\mathrm{5x\:-\:21\:=\:4}$

Let us add 21 to both sides of the equation

$$\mathrm{5x\:-\:21\:+\:21\:=\:4\:+\:21}$$

$$\mathrm{5x\:=\:25}$$

Now let us divide both sides of the equation by 5, we get

$$\mathrm{\frac{5x}{5}\:=\:\frac{25}{5}}$$

$$\mathrm{x\:=\:5}$$

One Equation One Variable

A linear equation with only one variable can be represented in the standard form as $\mathrm{Ax\:+\:c\:=\:0}$, Where x is variable, A is the coefficient of variable x, and c is constant.

For example, $\mathrm{8a\:+\:30\:=\:11}$

Two Equations Two Variables

A linear equation with only one variable can be represented in the standard form as

$$\mathrm{Ax\:+\:By\:+\:C\:=\:Px\:+\:Qy\:+\:R}$$

Where x and y are variables, A and B are the coefficients of variables x and y, respectively, and c is constant. For example

$$\mathrm{4x\:+\:7y\:+\:12\:=\:3x\:+\:4y\:+\:11}$$

Solved Examples

1)Adil’s age is twice that of Divya’s age. He was thrice Divya’s age 10 years ago. What are their present ages?

Answer − Let us assume Adil’s age be x and Divya’s age be y,

We know Adil’s age is twice that of Divya, then we can write

$$\mathrm{x\:=\:27\:\:\:\:\:\:\:Eq\:(1)}$$

10 years ago, Adil was thrice Divya’s age, we can write

$$\mathrm{x\:-\:10\:=\:3(y\:-\:10)}$$

On simplifying, we get

$$\mathrm{x\:-\:10\:=\:3y\:-\:30}$$

Putting the variables and constants on either side of the equation.

$$\mathrm{x\:-\:3y\:=\:-30\:+\:10}$$

$$\mathrm{x\:-\:3y\:=\:-20}$$

Multiplying both sides by -1

$$\mathrm{3y\:-\:x\:=\:20\:\:\:\:\:\:Eqn\:(2)}$$

Putting the value of x from Eqn (1) into Eqn (2)

$$\mathrm{3y\:-\:2y\:=\:20}$$

$$\mathrm{y\:=\:20}$$

Therefore, Divya’s age is 20.

Putting the value of y in Eqn (1).

$$\mathrm{x\:=\:2y}$$

$$\mathrm{x\:=\:40}$$

Therefore, Adil’s age is 40 years.

2)Two numbers when added give 50. The difference between the two numbers is 10. Find the two numbers with the help of linear equations.

Answer −

Let the number be a and b.

Then, we have two equations

$$\mathrm{a\:+\:b\:=\:50\:\:\:\:\:\:Eqn\:(1)}$$

$$\mathrm{a\:-\:b\:=\:10\:\:\:\:\:\:Eqn\:(2)}$$

Adding both the equations, we get

$$\mathrm{a\:+\:b\:+\:a\:-\:b\:=\:50\:+\:10}$$

$$\mathrm{2a\:=\:60}$$

$$\mathrm{a\:=\:30}$$

Putting the value of a in Eqn (1)

$$\mathrm{30\:+\:b\:=\:50}$$

$$\mathrm{b\:=\:20}$$

3)A bag contains coins of 25 paise and 50 paise worth Rs 20. If the bag contains a total of 56 coins. Find the number of each coin in the bag.

Answer − Let the number of 25 paise and 50 paise coins be x and y, respectively

Then we have two equations −

$$\mathrm{0.25x\:+\:0.50y\:=\:20}$$

On simplifying, we get

$$\mathrm{x\:+\:2\:=\:80\:\:\:\:\:Eqn\:(1)}$$

The bag has total 32 coins, so we can write

$$\mathrm{x\:+\:y\:=\:56\:\:\:\:\:Eqn\:(2)}$$

Subtracting Eqns (1) and (2)

$$\mathrm{x\:+\:2y\:-\:x\:-\:y\:=\:80\:-\:56}$$

$$\mathrm{y\:=\:24}$$

Putting the value of y in eqn (2), we get

$$\mathrm{x\:=\:32}$$

Therefore, the number of 25 paise coins is 32 and 50 paise coins is 24.

4)There are two whole numbers which differ by 28. The ratio between the numbers is 7:3. Find the numbers.

Answer − Let the two numbers be x and y

The ratio between two numbers is

$$\mathrm{\frac{x}{y}\:=\:\frac{7}{3}}$$

$$\mathrm{3x\:-\:7y\:=\:0\:\:\:\:\:Eqn\:(1)}$$

We also know that the difference between the numbers is 28

$$\mathrm{x\:-\:y\:=\:28}$$

Multiplying both sides by 3,

$$\mathrm{3x\:-\:3y\:=\:84\:\:\:\:\:Eqn\:(2)}$$

Subtracting Eqns (1) and (2),

$$\mathrm{3x\:-\:3y\:-\:3x\:+\:7y\:=\:84\:-\:0}$$

$$\mathrm{4y\:=\:84}$$

$$\mathrm{y\:=\:21}$$

Putting the value of y in equation $\mathrm{x\:-\:y\:=\:28}$

$$\mathrm{x\:=\:49}$$

Therefore the two numbers are 21 and 49.

5)A larger number is 4 less than 5 times of a smaller number. Whereas there sum is 38. Find the two numbers.

Answer − Let the larger number be x and smaller number be y.

We know tha the sum of the two numbrs is 38. Therefore, we can write

$$\mathrm{x\:+\:y\:=\:38\:\:\:\:\:\:Eqn\:(1)}$$

We also know that the larger number is 4 less than 5 times of a smaller number.

$$\mathrm{5y\:-\:x\:=\:4\:\:\:\:\:Eqn\:(2)}$$

Adding both the equations, we get

$$\mathrm{x\:+\:y\:+\:5y\:-\:x\:=\:38\:+\:4}$$

$$\mathrm{6y\:=\:42}$$

$$\mathrm{y\:=\:7}$$

Putting the value of y in Eqn (1), we get

$$\mathrm{x\:=\:31}$$

Therefore, we can say that the larger number is 31 and the smaller number is 7.

Conclusion

In this tutorial, we learned about Linear Equations, and applications of linear equations, algebra of linear equations, solving linear equations, one equation one variable and two variable equations. An equation may be defined as an algebraic expression equated to a constant or other expression.

A linear equation is also known as a one-degree equation. A linear equation is written in the form of $\mathrm{Ax\:+\:c\:=\:0}$ or $\mathrm{Ax\:+\:By\:+\:c\:=\:0}$. Where x and y are variables, A and B are the coefficients of variables x and y, respectively, and c is constant. An equation has two sides; left-hand side (LHS) and right-hand side (RHS). When a number or a variable is added or subtracted from both sides of an equation, the equation remains true.

FAQs

1. What are real life applications of linear equations?

Linear equations can help in so many ways in our real life, mostly for the prediction of values for the variables. For example, predicting profit over time, area enclosed straight lines, and calculating mileage.

2. In how many ways can one solve the linear equations?

Linear equations can be solved by substitution, elimination, cross multiplication, and graphical methods.

3. Can there be more than one solution to a linear equation?

No, there cannot be more than one solution to a linear equation. Although sometimes there can be no solution.

4. How many variables can a linear equation have?

A linear equation can have any number of variables depending on parameters.

5. What is the exponential power of a linear equation?

Exponential power of a linear equation is one.