The document Linear Equations - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev is a part of the Banking Exams Course Quantitative Aptitude for GMAT.

All you need of Banking Exams at this link: Banking Exams

**Introduction**

- Linear equations is one of the foundation topics in the Quant section of CAT.
- Hence, the fundamentals of this concept are useful in solving the questions of the other topics by assuming the unknown values as variables.
- Be careful of silly mistakes in this topic as that is how students generally lose marks here.
- Generally, the number of equations needed to solve the given problem is equal to the number of variables.

**What is a Linear Equation?**

A linear equation is an equation which gives straight line when plotted on

a graph.

- Linear equations can be of one variable or two variable or three variable.
- Let a, b, c and d are constants and x, y and z are variables. A general form of single variable linear equation is ax+b = 0.
- A general form of two variable linear equation is ax+by = c.
- A general form of three variable linear equation is ax+by+cz = d.

**➢ Equations with Two Variables**

- Consider two equations ax+by = c and mx+ny = p. Each of these equations represent two lines on the x-y co-ordinate plane. The solution of these equations is the point of intersection.
- If a / m = b / n ≠ c / p then the slope of the two equations is equal and so they are parallel to each other. Hence, no point of intersection occurs. Therefore no solution.
- If a / m ≠ b / n then the slope is different and so they intersect each other at a single point. Hence, it has a single solution.
- If a / m = b / n = c / p then the two lines are same and they have infinite points common to each other. So, infinite solutions occurs.

Try yourself:The pair of equations 3x – 5y = 7 and – 6x + 10y = 7 have:

View Solution

**➢ General Procedure to Solve Linear Equations**

- Aggregate the constant terms and variable terms.
- For equations with more than one variable, eliminate variables by substituting equations in their place.
- Hence, for two equations with two variables x and y, express y in terms of x and substitute this in the other equation.
**Example:**Let x + y = 14 and x + 4y = 26 then x = 14-y (from equation 1) substituting this in equation 2, we get 14 - y + 4y = 26. Hence, y = 4 and x = 10.- For equations of the form ax+by = c and mx+ny = p, find the LCM of b and n. Multiply each equation with a constant to make the y term coefficient equal to the LCM. Then subtract equation 2 from equation 1.
**Example:****Let 2x+3y = 13 and 3x+4y = 18 are the given equations (1) and (2).**

⇒ LCM of 3 and 4 is 12.

⇒ Multiplying (1) by 4 and (2) by 3, we get 8x+12y = 52 and 9x+12y = 54.

⇒ (2) - (1) gives x = 2, y = 3- If the system of equations has n variables with n-1 equations then the solution is indeterminate.
- If system of equations has n variables with n-1 equations with some additional conditions like the variables are integers then the solution may be determinate.
- If system of equations has n variables with n-1 equations then some combination of variables may be determinable.
**Example:**If ax+by+cz = d and mx+ny+pz = q, if a, b, c are in Arithmetic progression and m, n and p are in AP then the sum x+y+z is determinable.

**➢ Equations with Three Variables**

- Let the equations be a
_{1}x + b_{1}y + c_{1}z = d_{1,}a_{2}x+b_{2}y+c_{2}z = d_{2}and a_{3}x+b_{3}y+c_{3}z = d_{3}. 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 = a
_{1}(b_{2}c_{3}-c_{2}b_{3}) - b_{1}(a_{2}c_{3}- c_{2}a_{3}) + c_{1}(a_{2}b_{3}-b_{2}a_{3})

Offer running on EduRev: __Apply code STAYHOME200__ to get INR 200 off on our premium plan EduRev Infinity!

80 videos|99 docs|175 tests