How to Solve Linear Equations and Apply Them in Word Problems
Basic Mathematics is a part of one's daily life, and because of that, the introduction of Mathematics also takes place with real-life examples. Everyone has come across the age-old question "if you have 6 candies and give your friend 2, how many candies are left with you?". While with the advancement of classes, Maths becomes more about theorems and formulas, a part of it is always derived from practical experiences. The section, algebra deals with such real-world problems and algebra linear equations applications is one of the essential topics under it.
Linear equations real-world problems let a student formulate their equations to solve them. Every sum will require a new approach, though it is very possible to categorise certain situations under one type of equation. To understand this type of equation, one first needs to know different aspects of it.
Linear Equations
A linear equation, according to Mathematics, is an equation of first-order that is defined for lines in a coordinate system.
The most common form of a linear equation is,
y= ax + b
Where a is the slope of the line and b is the y-intercept.
This can also help one to understand the relation between two variables that is y and x. Different forms of this equation can be used to solve different types of linear algebra real-world problems. Let's solve a real-world problem to understand how one can formulate a linear equation and solve it.
Q1: Raj went to buy flowers pots in his local nursery. The shopkeeper said that there are two types of pots, terracotta and plastic. He also said that the price of a terracotta pot is three times the price of a plastic pot. If Raj can buy one of each at 100 rupees, what is the price of each pot?
Ans: First, understand the relation between the two costs.
Now, let's assume that the cost of a plastic pot is x, then the cost of a terracotta pot will be three times x that id 3x.
It is stated that one of each pot can be bought using 100 rupees, that means
x + 3x = 100, this is the linear equation for this given problem.
Now solve it for x, that is
4x = 100
x = 100/4
x = 20
So, the price of a plastic pot is 25 rupees. Since a terracotta pot costs three times, its price will be 3 × 25 = 75 rupees.
Under such circumstances where one only needs to find one variable, solving word problems with linear equations are usually like the example above. But if one needs to find the value of more than one variable, a system of linear equations is used.
Applying Systems of Linear Equations
Linear system or system of linear equations is a set of one or more linear equations that involve a common set of variables. For example,
3x + 2y = 3
-2x + y = 4
These two equations are together a system of linear equations. Algebra linear equations applications are not much different from that of solving for a single variable, one has to formulate their own equations here as well. This can be better understood by practising real-life linear equations word problems. Let's solve one such problem to see the course of equation formulation.
Q2: The cost of passes for a theme park is 162 rupees for 12 children and 3 adults. For the same theme park, the cost of passes for 8 children and 3 adults was 122 rupees. What is the cost of one pass for each child and adult?
Ans: From the question, it is clear that these two situations will translate into two equations. Now, put your understanding into these two situations and form equations just the way it was done above. System of equations real-world problems can have multiple variables; in this case, there are two children and adults.
Let the cost of a child's pass be x, and the cost of an adult's pass be y. So, according to the first situation,
12x + 3y = 162
And according to the second situation
8x + 3y = 122
Now, subtract the first equation from the second one, like
12x + 3y = 162
- 8x + 3y =122
4x = 40
x = 40/4
x = 10
Put the value of x in any of the equations to y’s value, like
8(10) + 3y = 122
80 + 3y = 122
3y = 42
y = 14
So, the cost of a child’s pass is 10 rupees and 14 rupees for an adult’s pass.
Algebra linear equations applications simplify relations between two or some times more variables. Yet, there are several aspects that a student should understand before solving its questions. Vedantu provides live coaching classes on a variety of mathematical topics including linear equations and its different applications. Check their sample sheets to practice more real-world problems and master the skill of equation formulation.
FAQs on Algebra Linear Equations and Their Real Life Applications
1. What is a linear equation in algebra?
A linear equation is an equation in which the highest power of the variable is 1. It can be written in the form ax + b = 0, where a ≠ 0.
- It represents a straight line when graphed.
- It can have one variable (e.g., 2x + 3 = 7) or two variables (e.g., 2x + y = 5).
- The solution is the value that makes the equation true.
2. How do you solve a linear equation step by step?
To solve a linear equation, isolate the variable using inverse operations. For example, solve 3x − 5 = 10:
- Add 5 to both sides: 3x = 15
- Divide by 3: x = 5
3. What is the standard form of a linear equation?
The standard form of a linear equation in two variables is Ax + By = C, where A, B, and C are constants and A and B are not both zero.
- A, B, and C are usually integers.
- It represents a straight line on a coordinate plane.
- Example: 2x + 3y = 6.
4. What are the real-life applications of linear equations?
Linear equations are used to model relationships with a constant rate of change in real life. Common applications include:
- Cost calculations: Total cost = fixed cost + (price × quantity)
- Distance problems: Distance = speed × time
- Budget planning and financial forecasting
- Supply and demand analysis in economics
5. How do you solve a word problem using linear equations?
To solve a word problem using a linear equation, translate the situation into an equation and solve for the unknown. Steps:
- Define the variable (e.g., let x be the number).
- Write an equation based on the problem.
- Solve using algebraic steps.
6. What is the slope-intercept form of a linear equation?
The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
- m represents the rate of change.
- b is the value of y when x = 0.
- Example: In y = 2x + 3, slope = 2 and y-intercept = 3.
7. What is the difference between a linear equation in one variable and two variables?
A linear equation in one variable has a single solution, while a linear equation in two variables has infinitely many solutions.
- One variable example: 2x + 3 = 7 (solution: x = 2).
- Two variables example: 2x + y = 5 (solutions are ordered pairs like (1,3)).
- Graphically, one variable gives a point, two variables give a straight line.
8. How do you graph a linear equation?
To graph a linear equation, plot at least two points that satisfy the equation and draw a straight line through them. Example for y = x + 1:
- If x = 0, then y = 1 → point (0,1)
- If x = 1, then y = 2 → point (1,2)
- Draw a straight line through these points.
9. What are common mistakes when solving linear equations?
Common mistakes when solving linear equations include incorrect sign changes and unequal operations on both sides.
- Forgetting to apply operations to both sides of the equation.
- Errors when distributing negative signs.
- Miscalculating during simplification.
10. Can you give an example of solving a linear equation in real life?
Yes, linear equations can model cost problems such as calculating total taxi fare. Example: A taxi charges $3 base fare plus $2 per mile, so the equation is C = 2x + 3.
- If you travel 5 miles: C = 2(5) + 3
- C = 10 + 3
- Total cost = $13
