Important Concepts: Linear Equations - CAT PDF Download

What is a Linear Equation?

A linear equation represents a straight line when plotted on a graph. 
It involves variables with a degree of 1 (no exponents or roots) and can have one, two, or three variables. For CAT, linear equations in two variables are most common, often appearing in algebra, geometry, or word problems.

General Forms

• One Variable: ax + b = 0 (e.g., 2x - 4 = 0)

• Two Variables: ax + by = c (e.g., 3x + 4y = 12)

• Three Variables: ax + by + cz = d (less common in CAT but useful for advanced problems)

Linear Equations in Two Variables

A linear equation in two variables is of the form ax + by + c = 0, where:

• x and y are variables.
• a, b, and c are real numbers, with a and b non-zero.

These equations are called simultaneous linear equations when paired (e.g., solving two equations together to find x and y). They are used to find coordinates of a point, slopes of lines, or solutions to real-world problems like cost-profit analysis or distance-time problems in CAT.

Forms of Linear Equations in Two Variables

Linear equations in two variables are equations with a unique solution, no solutions, or infinitely many solutions. They can be present in different forms:

1. Standard form
2. Intercept form
3. Point-slope form

1. Standard Form of Linear Equations in Two Variables

• The format of the equation representation in standard form is ax + by + c = 0
• Here a, b and c are the coefficient constants.
• Consider an equation as 3x + 4y = 11. In standard form, the equation is represented as:
  3x + 4y - 11 = 0

2. Intercept Form of Linear Equations in Two Variables

• The format of the equation representation in intercept form is: y = mx + b
• In the above equation, m is the slope and b stands for the y-intercept.
• Consider the same equation as 3x + 4y = 11. In the intercept or say slope-intercept form for the equation is represented as:
  y = (-3/4)x + 11/4

3. Point Slope Form of Linear Equations in Two Variables

• The format of the equation representation in point-slope form is: y − y₁ = m(x − x₁)
• Where m=slope and (x₁, y₁) denotes a point on the given line.
• An example of the point-slope form is: y - 4 = 5(x - 3)
  Here 5 is the slope and (3, 4) denotes a point on the given line.

Nature of Solution of Linear Equations in Two Variables

• A system of linear equations in two variables can have one solution or no solution or infinitely many solutions. 
• A system of linear equations is consistent if it has a solution, meaning that there is at least one set of values that satisfies all the equations. On the other hand, an inconsistent system does not have a solution, indicating that no set of values can satisfy all the equations simultaneously.
• Similarly, a system of linear equations is considered independent if it has a unique solution, meaning that the values of the variables can be determined uniquely. In contrast, a dependent system has an infinite number of solutions.

1. Unique Solution of Linear Equations in Two Variables

• A system of linear equations in two variables has a unique solution if and only if the equations represent two non-parallel lines that intersect at exactly one point. In other words, the lines must have different slopes, indicating that they are not parallel, and they must intersect at one point only. 
• Therefore, if m₁ and m₂ are the slopes of the equations of the two lines, then m₁ should not be equal to m₂ in order for the equations to have a unique solution, that is, m₁ ≠ m₂.
• The unique solution can be found by solving the equations simultaneously using various methods, including substitution, elimination, or matrices. It is important to note that a system of linear equations may have zero solutions or infinitely many solutions if the lines are parallel or coincide with each other.

2. No Solution of Linear Equations in Two Variables

• A system of linear equations in two variables has no solution if and only if the equations represent two parallel lines that do not intersect. In other words, the lines must have the same slope, indicating that they are parallel, and they do not intersect.
• This can be represented mathematically as m₁ = m₂, where m₁ and m₂ are the slopes of the lines.
• When a system of linear equations has no solution, it means that there is no value for each variable that satisfies both equations simultaneously. This situation is often called an inconsistent system, and it arises when the lines represented by the equations do not cross each other.

3. Infinite Solutions of Linear Equations in Two Variables

• It is important to note that a system of linear equations may have infinitely many solutions if the equations represent the same line or coincident lines. 
• In such a case on of the linear equation is a scalar multiple of the other equation, like in the case of x + y =2 and 2x + 2y = 4.

How to Solve Linear Equations in Two Variables

The various methods to solve linear equations in two variables are given below:

1. Substitution Method
2. Elimination Method
3. Cross-multiplication method
4. Graphical Method
5. Determinant Method

1. Substitution Method to Solve Linear Equations in Two Variables

The procedure of Substitution Method to Solve Linear Equations in Two Variables is as follows:
Step 1. Solve one of the given equations to get the value of one of the variables in terms of the other, whichever is convenient.
Step 2. Substitute the value of the variable so obtained in the other equation.
Step 3. Solve the resulting single variable equation. Now substitute this value into either of the two original equations and solve it to find the value of the second variable.

Example 1. Solve the following system of linear equations:
4x-3y = 8
x-2y = -3

Sol: The given equations are
4x-3y=8 ……….(i)
x-2y= -3 ……….(ii)
We can solve either equation for either variable. But to avoid fractions, we solve the second equation for x,
x = 2y - 3 ……….(iii)
Substituting this value of x in equation (i), we get
4(2y-3)-3y=8
8y-12-3y=8
5y=20
y=4.
Substituting this value of y in (ii), we get
x-24= -3
x-8= -3
x=5.
Hence, the solution is x = 5, y = 4.

Example 2. Solve the following system of linear equations:
8x+5y=9
3x+2y=4.

Sol: The given equations are
8x+5y=9 ……….(i)
3x+2y=4 ……….(ii)
From equation (ii), we get
2y = 4−3xy

Substituting this value of y in (i), we get

Substituting this value of x in equation (ii), we get
3(-2) + 2y = 4
2y = 10
y = 5.
Hence, the solution is x = -2, y = 5.

2. Elimination Method to Solve Linear Equations in Two Variables

Steps:
1. Multiply equations to make the coefficients of one variable equal.
2. Add or subtract to eliminate one variable.
3. Solve for the remaining variable, then back-substitute.

Example: 
Solve:
8x + 5y = 9
3x + 2y = 4

Sol: Multiply the second equation by 5 and the first by 2 to align y-coefficients:
16x + 10y = 18
15x + 10y = 20
Subtract: 
(16x + 10y) - (15x + 10y) 
= 18 - 20 
 x = -2
Substitute x = -2 into 3x + 2y = 4: 
3(-2) + 2y = 4 
 2y = 10 
 y = 5
(x, y) = (-2, 5)

3. Cross-Multiplication Method to Solve Linear Equations in Two Variables

Procedure of Cross-Multiplication Method to Solve Linear Equations in Two Variables: 

Let the system of simultaneous linear equations be
a₁x + b₁y + c₁=0
a₂x + b₂y + c₂ = 0
To solve this system of linear equations by cross-multiplication method, the solution is given by
where, 

4. Graphical Method to Solve Linear Equations in Two Variables

The next method to solve linear equations in two variables is the graphical approach. To decode two linear equations in two variables graphically we will follow the below steps:

• Step 1: We will start with Plotting the two equations on the graph.
• Step 2: To plot the graph manually, transform the equations to the form y = mx + b or x = my  + b.
• Step 3: Substitute the different values of x like 0, 1, 2,…… and obtain the related values of y, or vice-versa to obtain different values of x.
• Step 4: Plot the different points of the equation on the graph and try to locate the point where both lines intersect one another.
• Step 5: The point of the meeting is the answer to the given system of equations.

It is not always possible that both the lines will intersect one another, they can even be parallel or coincide with each other. In such a case we can follow the below conclusions:

If we are given a system of two linear equations: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0

• If ᠎a₁/a₂ ≠ b₁/b₂
  In such a case a unique solution is obtained also the given set of lines intersects at a single point.
• If a₁/a₂ = b₁/b₂ ≠ c₁/c₂
  In such a case the system equations have no solution also the lines are parallel to one another.
• a₁/a₂ = b₁/b₂ = c₁/c₂
  For the above case, the system equations have an infinite number of solutions and the given two lines coincide with each other. If the system holds a solution, then it is stated to be consistent; otherwise, it is considered inconsistent.

5. Determinant Method to Solve Linear Equations in Two Variables

Under this method, we will learn to determine the solution for a system of linear equations in two variables. The steps are as follows:
Step 1: Consider the questions as: a₁x + b₁y = c₁ and a₂x + b₂y = c₂
Step 2: We would first locate the determinant developed by the coefficients of x and y and mark it as Δ.

Step 3: Next we will obtain the determinant Δx which is the determinant calculated by replacing the first column of Δ with the constant terms in the equation.

Step 4: Similarly we will determine the determinant Δy which is calculated by replacing the second column of Δ with the constant terms in the equation.

Step 5: Lastly the solution for the provided system of linear equations is received by the formulas:

Equations with Three Variables

• Let the equations be a₁x + b₁y + c₁z = d_1, a₂x+b₂y+c₂z = d₂ and a₃x+b₃y+c₃z = d₃ . Here we define the following matrices:
• If Determinant of D ≠ 0, then the equations have a unique solution. 
• If Determinant of D = 0, and at least one but not all of the determinants Dx , Dy or Dz is zero, then no solution exists. 
• If Determinant of D = 0, and all the three of the determinants Dx , Dy and Dz are zero, then there are infinitely many solution exists. 
• Determinant can be calculated by D = a₁(b₂c₃ -c₂b₃) - b₁(a₂c₃ - c₂a₃) + c₁(a₂b₃ -b₂a₃)

Solving Equations with Three Variables

Though less common in CAT, three-variable equations (a₁x + b₁y + c₁z = d₁, etc.) may appear in complex word problems. 
Solve using:

• Elimination: Eliminate one variable by combining pairs of equations, then solve the resulting two-variable system.

• Determinant Method: Use Cramer’s Rule with 3x3 matrices if the determinant D ≠ 0.

Example: 
Solve:
x + y + z = 2
2x - y + z = 3
x + 2y - z = 1

Sol: Add the first two equations: 3x + 2z = 5
Add the first and third: 2x + 3y = 3
Solve the new system: x = 1, y = -1, z = 2
(x, y, z) = (1, -1, 2)

Solved Examples

Q1: For some real numbers a and b, the system of equations   has infinitely many solutions for x and y. Then, the maximum possible value of ab is:
(a) 33
(b) 25
(c) 15
(d) 55
Ans: (a)

Sol: 

Q2: Three friends, returning from a movie, stopped to eat at a restaurant. After dinner, they paid their bill and noticed a bowl of mints at the front counter. Sita took one-third of the mints, but returned four because she had a momentary pang of guilt. Fatima then took one-fourth of what was left but returned three for similar reason. Eswari then took half of the remainder but threw two back into the bowl. The bowl had only 17 mints left when the raid was over. How many mints were originally in the bowl?
(a) 38
(b) 31
(c) 41
(d) None of these
 Answer (D)

Solution:

Q3:  In 2010, a library contained a total of 11500 books in two categories – fiction and nonfiction. In 2015, the library contained a total of 12760 books in these two categories. During this period, there was 10% increase in the fiction category while there was 12% increase in the non-fiction category. How many fiction books were in the library in 2015?
(a) 6160
(b) 6600
(c) 6000
(d) 5500
Answer (B)

Solution:

Tips to Avoid Common Mistakes 

1. Check Coefficients Carefully: Misreading coefficients (e.g., 3x as 2x) is a common error. Double-check numbers before solving.

2. Verify Solutions: Substitute answers back into both equations to confirm correctness, especially in substitution or elimination methods.

3. Avoid Fraction Errors: Simplify fractions correctly when using substitution or cross-multiplication to prevent calculation mistakes.

4. Understand Solution Conditions: Before solving, check if the system has a unique, no, or infinite solutions using a₁/a₂, b₁/b₂, c₁/c₂ ratios to4. save time.

5. Time Management: For CAT, prefer elimination or substitution over graphical methods due to time constraints. Practice quick mental math for coefficients.

6. Word Problem Clarity: Translate word problems into equations carefully. 
Define variables clearly (e.g., let x = number of fiction books) to avoid misinterpretation.

7. Three-Variable Systems: Break them into two-variable systems systematically to avoid confusion. Always verify by substituting into all equations.