How to Solve Linear Equations Step by Step
The concept of linear equations plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding linear equations is essential for algebra, problem-solving, and reasoning skills used in daily life and competitive exams.
What Is Linear Equation?
A linear equation is an equation where the highest power of each variable is one. You’ll find this concept applied in areas such as algebraic expressions, word problems, and graphical analysis. In a linear equation, variables like x or y are never raised to powers higher than one, and the solution always produces a straight line when graphed.
Key Formula for Linear Equation
Here’s the standard formula: \( ax + b = 0 \), where a and b are constants and x is the unknown variable.
For two variables (common in higher classes): \( ax + by + c = 0 \)
Cross-Disciplinary Usage
Linear equations are not only useful in Maths but also play an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in various questions, such as problems involving speed, electricity, data analysis, and much more. At Vedantu, learning how to set up and solve a linear equation forms the foundation for all future math topics and logical thinking.
Linear Equation Examples
Let’s look at a few examples of linear equations:
- Linear equation in one variable: 3x + 4 = 10
- Linear equation with fractions: (2x/3) - 5 = 7
- Linear equation in two variables: 2x + 3y = 12
- Word problem example: "The sum of a number and seven is 15. What is the number?" → x + 7 = 15
Step-by-Step Illustration
- Start with the given: \( 3x + 5 = 20 \)
Subtract 5 from both sides: \( 3x = 15 \)
- Divide by 3:
\( x = 5 \)
How to Solve Linear Equations with Fractions
When dealing with fractions, clear the fractions first to make calculations easy.
1. Start with the equation\( \frac{2x}{3} - 5 = 7 \)
2. Add 5 to both sides
\( \frac{2x}{3} = 12 \)
3. Multiply both sides by 3
\( 2x = 36 \)
4. Divide by 2
\( x = 18 \)
Linear Equations in Two Variables
| Type | General Form | Example |
|---|---|---|
| One Variable | ax + b = 0 | 2x + 5 = 9 |
| Two Variables | ax + by + c = 0 | x + 2y - 7 = 0 |
In two variables, you usually solve for both x and y using methods like substitution, elimination, or by plotting graphs.
Graphical Representation
A linear equation in two variables can be plotted as a straight line on a graph. For example, \( y = 2x + 1 \) passes through all points where for every x, y equals twice the x value, plus one. Both Linear Graphs and plotting tools can help you visualize solutions and check your answers easily.
Speed Trick or Vedic Shortcut
Here’s a quick way to check if your solution is correct in exams: substitute your answer back into the original linear equation to confirm both sides are equal. For equations with two variables, try plugging in both values to see if the statement is balanced. This ‘reverse check’ helps you avoid calculation mistakes.
Example Trick: For ax + b = 0, just isolate x directly if possible—no need to rearrange many steps. Practice this, and you’ll save time during tests.
Try These Yourself
- Solve: \( 2x - 3 = 11 \)
- If \( 5x + 6 = 21 \), what is \( x \)?
- Solve the system: \( x + y = 10 \) and \( x - y = 4 \)
- Frame a linear equation from: "The sum of a number and 12 is 35."
Frequent Errors and Misunderstandings
- Forgetting to apply the same operation to both sides of the equation.
- Incorrectly combining like terms.
- Missing sign changes when moving terms across the ‘=’ sign.
- Confusing linear equations with quadratic or higher-degree equations.
Relation to Other Concepts
The idea of linear equations connects closely with topics such as algebraic equations and systems of equations. Mastering these helps understand graphs, inequalities, and more advanced maths topics covered in later classes and competitive exams.
Classroom Tip
A quick way to remember linear equations is “line = linear = straight lines on a graph.” All rules for balancing scales in real life work the same when solving equations: whatever you do to one side, do to the other. Vedantu’s teachers often use these real-world analogies and live quizzes to make learning easy and fun.
We explored linear equations—from definition, formula, examples, frequent mistakes, and their connection to other maths and science topics. Continue practicing with Vedantu’s linear equation worksheets to become confident in solving any equation you see in homework or exams!
Also explore: Linear Equations in One Variable, Linear Equations in Two Variables, Algebra, for quick answers.
FAQs on Linear Equations Explained: Concepts, Solving Tips & Practice
1. What is a linear equation in Maths?
A linear equation is an algebraic equation where the highest power of the variable is one. It represents a straight line when graphed. The general form for a linear equation in one variable is ax + b = 0, where 'a' and 'b' are constants, and 'x' is the variable. For two variables, the general form is ax + by + c = 0, where 'a', 'b', and 'c' are constants, and 'x' and 'y' are the variables.
2. How do you solve a linear equation step by step?
Solving a linear equation involves isolating the variable. Here's a step-by-step guide:
- Simplify both sides: Combine like terms and remove parentheses.
- Move variable terms to one side: Add or subtract terms to get all terms with the variable on one side of the equation and constant terms on the other.
- Isolate the variable: Use inverse operations (addition/subtraction, multiplication/division) to solve for the variable.
- Check your solution: Substitute the value back into the original equation to verify it's correct.
3. Can you give examples of linear equations in one and two variables?
One variable: 2x + 5 = 11; 3y - 7 = 2; x/2 = 4 Two variables: 3x + y = 7; 2x - 4y = 6; x + 2y = 10
4. Is y = -7 a linear equation?
Yes, y = -7 is a linear equation. It represents a horizontal line at y = -7. Although it only has one variable, it still fits the definition of a linear equation because the highest power of the variable is one (y can be thought of as y1).
5. What is the formula to write a linear equation?
The formula depends on the form you want to use:
- Standard form (one variable): ax + b = 0
- Standard form (two variables): ax + by + c = 0
- Slope-intercept form: y = mx + c (where m is the slope and c is the y-intercept)
- Point-slope form: y - y1 = m(x - x1) (where m is the slope and (x1, y1) is a point on the line)
6. How are linear equations used in real-life situations?
Linear equations have many real-world applications, such as:
- Calculating costs: Determining the total cost based on a fixed cost and a variable cost per unit.
- Predicting values: Estimating future values based on a trend shown by past data.
- Analyzing relationships: Modeling the relationships between variables in many scientific or engineering fields.
- Solving mixture problems: Finding the amount of two solutions needed to obtain a desired concentration.
7. How can you identify if an equation is not linear?
An equation is not linear if it contains:
- Variables raised to powers other than 1 (e.g., x², y³).
- Products of variables (e.g., xy).
- Variables in the denominator (e.g., 1/x).
- Variables inside radicals or absolute values.
8. What are the differences between solving linear and quadratic equations?
Linear equations have a highest power of 1 and typically have one solution. They are solved using basic algebraic manipulations. Quadratic equations have a highest power of 2 and may have up to two solutions. They are often solved using methods like factoring, the quadratic formula, or completing the square.
9. How do linear equations appear in graphs, and what does the slope represent?
Linear equations always graph as straight lines. The slope of the line (represented by 'm' in y = mx + c) indicates the steepness and direction of the line. A positive slope means the line goes upward from left to right; a negative slope means it goes downward. A slope of 0 indicates a horizontal line, and an undefined slope indicates a vertical line.
10. What are some common mistakes to avoid when solving linear equations?
Common mistakes include:
- Incorrectly applying the distributive property.
- Making errors when working with negative signs.
- Forgetting to perform the same operation on both sides of the equation.
- Dividing by zero.
- Incorrectly combining like terms.
11. What is the difference between linear equations in one variable and linear equations in two variables?
A linear equation in one variable has only one variable (e.g., 2x + 5 = 9), and its solution is a single value for that variable. A linear equation in two variables has two variables (e.g., 3x + y = 7), and its solution is a set of ordered pairs (x, y) that satisfy the equation. These solutions form a straight line on a graph.
